A  "BROKEN"  TRANSIT. 
Length  of  axis,   34  inches.     Approximate  cost,  $1200. 


A  TEXT-BOOK 


OF 


FIELD   ASTRONOMY 


s. 


GEO 

Director  of  the  Wa$JH>\irn  Observatory, 
Professor  of  Astronomy  in  the  University  of  Wisconsin* 


SECOND  EDITION,   REVISED   AND   ENLARGED. 
FIRST  THOUSAND.^    "  ?, 


NEW  YORK : 

JOHN  WILEY  &  SONS. 

LONDON  :  CHAPMAN  &  HALL,  LIMITED. 

1912 


Astron.  Dept. 


Copyright,  1902,  1908, 

BY 
GEORGE  C.  COMSTOCK. 


THE  SCIENTIFIC   PRESS 

ROBERT   DRUMMOND   AND    COMPANY 

BROOKLYN,    N.    Y. 


PREFACE  TO  THE  SECOND  EDITION. 


THE  author  has  embraced  the  opportunity  afforded 
by  a  second  edition  of  this  work  to  introduce  into  it 
certain  changes  which,  without  modifying  the  general 
plan  and  scope,  will,  it  is  hoped,  increase  its  usefulness. 
For  the  most  part  these  changes  are  made  in  the  direc- 
tion of  increased  simplicity  or  of  increased  precision, 
and  are  most  conspicuously  shown  in  §§  19,  23,  3,2,  37 
and  38.  The  tables  at  the  end  of  the  book  have  been 
considerably  increased,  both  in  extent  and  precision,  and 
now  suffice  for  approximate  as  well  as  rough  deter- 
minations of  time,  latitude  and  azimuth  without  the 
use  of  an  almanac. 

June  i,  1908. 
March  i,  1912. 


670910 


PREFACE  TO  THE  FIRST  EDITION. 


THE  present  work  is  not  designed  for  professional 
students  of  astronomy,  but  for  another  and  larger  class 
found  in  technical  colleges.  For  many  years  it  has  been 
the  author's  duty  to  teach  to  students  of  engineering 
the  elements  of  practical  astronomy,  and  the  experience 
thus  acquired  has  gradually  produced  the  unconventional 
views  that  find  expression  in  the  present  text  and  which, 
to  the  author's  mind,  are  justified  by  the  following 
considerations : 

In  the  engineering  curriculum,  work  in  astronomy  is 
a  part  of  a  course  of  technical  and  professional  training 
of  students  who  have  no  purpose  to  become  astronomers. 
Under  these  circumstances  it  seems  the  duty  of  the 
instructor  to  select  for  presentation  those  parts  of 
astronomical  practice  most  closely  related  to  the  work 
of  the  future  engineer  and,  with  reference  to  the  narrow 
limits  of  time  allotted  the  subject,  to  keep  in  the  back- 
ground many  collateral  matters  that  are  of  primary  in- 
terest and  importance  to  the  student  of  astronomy  as  a 
science. 

The  parts  of  astronomical  practice  most  pertinent  to 


vi  PREFACE. 

engineering  instruction  seem  to  the  author  to  be  (a)  Train- 
ing in  the  art  of  numerical  computation ;  (6)  Training  in 
the  accurate  use  of  such  typical  instruments  of  precision 
as  the  sextant  and  the  theodolite,  with  special  refer- 
ence to  the  elimination  of  their  errors  from  the  results  of 
observation;  (c)  Determinations  of  time,  latitude,  and 
azimuth,  with  portable  instruments,  as  furnishing  sub- 
ject-matter through  which  a  and  b  may  be  conveniently 
realized.  If  this  work  is  to  be  done  during  the  single 
semester  usually  allowed  for  the  subject,  the  time  given 
to  its  theoretical  side,  spherical  astronomy,  must  be 
reduced  to  the  minimum  amount  compatible  with  the 
student's  intelligent  use  of  his  apparatus  and  formulas, 
and  in  the  present  work  this  pruning  of  the  theoretical  side 
has  been  carried  to  an  extent  that  would  be  unpardon- 
able in  the  training  of  an  astronomer,  but  which  appears 
necessary  and  proper  in  this  case. 

Since  many  engineering  students  acquire  from  the 
mathematical  curriculum  little  or  no  knowledge  of 
spherical  trigonometry  and  its  numerical  applications, 
the  first  chapter  of  the  work  is  devoted  to  a  brief  presen- 
tation of  the  elements  of  this  subject  with  special  refer- 
ence to  its  astronomical  uses  and  to  the  student's  acqui- 
sition of  good  habits  in  the  conduct  of  numerical  work. 
The  astronomical  problems  presented  in  the  following 
chapters  are  those  that  have  been  indicated  by  experience 
as  best  adapted  to  the  author's  own  pupils,  and  while 
many  of  the  methods  given  for  their  solution  are  not 
contained  in  the  current  text-books,  in  every  case  these 
are  either  methods  in  use  in  the  best  geodetic  surveys, 


PREFACE.  vii 

or  such  as  have  been  repeatedly  tested  with  students  and 
found  well  suited  to  their  use.  These  methods  are  classi- 
fied in  the  text  as  rough,  approximate,  and  precise,  with 
respect  to  their  precision  and  the  corresponding  amount 
of  time  and  labor  required  for  their  application,  and  the 
student  is  advised  not  to  use  the  refined  and  laborious 
methods  when  only  a  rough  result  is  required. 

As  a  rule,  in  the  development  of  formulae  no  attempt 
has  been  made  to  deal  with  the  general  case  when  the 
solution  of  a  particular  case  would  suffice  for  the  prob- 
lem in  hand;  e.g.,  the  earth's  compression  is  ignored 
in  treating  of  the  effect  of  parallax,  since  its  influence 
is  vanishingly  small  in  the  great  majority  of  cases  that 
the  student  will  ever  encounter,  and  cases  in  which  this 
influence  is  of  sensible  amount  should  be  avoided  by  the 
instructor.  A  more  serious  omission,  but  one  required 
by  the  general  plan  of  the  work,  is  found  in  the  theory 
of  the  transit  instrument,  Chapter  IX,  where  broken 
transits,  thread  intervals,  curvature  of  a  star's  apparent 
path,  flexure,  etc.,  are  passed  by  without  treatment  or 
even  suggestion.  They  are  not  required  for  the  begin- 
nings of  work  with  a  transit  instrument,  and  therefore 
constitute  a  part  of  more  advanced  study  than  is  here 
contemplated.  As  a  partial  guide  to  such  study  there 
is  given  upon  a  subsequent  page  a  list  of  references  to 
works  that  may  be  consulted  with  profit  by  the  student 
who  seeks  a  more  complete  knowledge  of  the  processes 
of  practical  astronomy. 

The  adopted  notation  follows,  with  only  slight  devia- 
tions, that  of  Chauvenet,  to  whose  elaborate  treatise 


vill  PREFACE. 

upon  Spherical  and  Practical  Astronomy  the  author  is 
under  obligations  that  are  common  to  every  present-day 
writer  upon  those  subjects.  His  thanks  are  also  due 
to  many  of  his  former  pupils,  and  in  particular  to  Dr. 
'S.  D.  Townley  and  Dr.  Joel  Stebbins,  who  have  read 
and  criticised  portions  of  his  manuscript. 

It  is  also  a  pleasure  to  acknowledge  here  the  comment 
and  criticism  that  has  come  to  the  author  from  practis- 
ing engineers  who  have  made  use  of  the  work.  In  part 
these  suggestions  are  incorporated  in  the  present  text, 
and  in  no  case  have  they  been  rejected  without  careful 
consideration. 


TABLE  OF  SYMBOLS. 


THE  following  table  contains  a  brief  explanation  of  the 
principal  symbols  employed  in  the  text,  with  references 
to  the  page  at  which  they  are  respectively  defined.  There 
are  omitted  from  the  table  a  considerable  number  of 
symbols  employed  only  in  immediate  connection  with 
their  definition. 

Mathematical. 

2  p.  154     Summation  symbol. 

[  ]  ii  The  enclosed  number  is  a  logarithm. 

Coordinates,  etc. 

h  p.    26  Altitude. 

e  25  Zenith  distance.     Complement  of  h. 

A  26  Azimuth,  reckoned  from  south. 

a0,  d0  65  Azimuth,  reckoned  from  north. 

t  26  Hour  angle. 

3  26  Declination. 

a  26  Right  ascension. 

<f>  29  Latitude. 

A  42  Longitude. 

5  52  Sun's  semi-diameter. 
P  53  Horizontal  parallax. 
R,  R'  58  Refraction,  etc. 

p  70     Polar  distance,  =  oo°— 5. 

Time. 

6  p.    30     Sidereal  time. 
M  41     Mean  solar  time. 

T  44  Time  shown  by  a  chronometer,  whether  right  or  wrong, 

AT  44  Chronometer  correction. 

p  45  Chronometer  rate. 

E  40  Equation  of  time. 

V  40  Date  of  conjunction,  mean  sun  with  vernal  equinox. 


X  TABLE  OF  SYMBOLS. 

Q  p.    42     Sidereal  time  of  mean  noon,  as  given  in  almanac. 

Qv  43     Sidereal  time  of  mean  noon  reduced  to  the  local  merid- 

ian. 

Rough  and  Approximate  Determinations. 

a0  p.     71     Tabular  difference  of  azimuth,  Polaris  and  north  pole. 

60  71     Tabular  difference  of  altitude,  Polaris  and  north  pole. 

F,f  71     Factors  to  transform  a0  and  b0  into  their  true  local 

values, 

H  87     A  horizontal  angle. 

y,  D  72      Auxiliaries  for  finding  hour  angle  of  Polaris. 

U  91     An  approximate  value  of  the  chronometer  correction, 

AT,  referred  to  sidereal  time. 

A9  An  approximate  azimuth  of  a  mark. 

x,  y  91     Corrections  to  transform  U  and  A9  into  true  values. 

dA 
C  92     A  coefficient,  equals  -T-. 

r  92     An  hour  angle. 

Instruments. 

R,  r         p.  118     Circle  or  micrometer  readings. 

d  100     Value  of  half  a  level  division. 

k  159     Value  of  one  revolution  of  a  micrometer. 

f  118     Deviation  of  a  vertical  axis  from  the  true  vertical. 

6',  6"  114     Components  of  f  parallel  and  perpendicular  to  hori- 

zontal axis. 

*  118     Deviation,  from  90°,  of  angle  between  axes  of  theodo- 

lite. 

w  1 20     Complement  of  spherical  angle  at  zenith  between  hori- 

zontal axis  and  line  of  sight. 

a  171     Deviation  of  horizontal  axis  from  true  east  and  west. 

b  171     Deviation  of  horizontal  axis  from  true  leveL 

c  171     Collimation  constant. 

A,B,C       173     Mayer's  transit  factors. 

Cm  162     Calibration  correction  to  micrometer-screw. 

Precise  Determinations. 

L  p.  144     Correction  to  equal  altitudes.      (Time.) 

ARQ  148     Correction  to  equal  altitudes.      (Azimuth.) 

g,  k  149     Auxiliaries  used  in  computation  of  azimuths. 

s  1 60     Auxiliary  used  in  computation  of  differential  refrac- 

tion. 

/  83     Auxiliary  used  in  computation  of  reduction  to  merid- 

ian. 
a  189     Auxiliary  collimation  coefficient. 


TABLE  OF  CONTENTS. 


CHAPTER  I. 

PAGB 

INTRODUCTORY i 

Spherical  trigonometry.    Approximate  formulas.    Numeri- 
cal computations.     Logarithmic  tables.     Limits  of  accuracy. 


CHAPTER  II. 
COORDINATES 22 

Fundamental  concepts.  Definitions.  Notation.  Table  of 
coordinates.  Transformation  of  coordinates. 

CHAPTER  III. 
TIME 35 

Three  time  systems.  Longitude.  Conversion  of  time. 
Chronometer  corrections.  The  almanac. 

CHAPTER  IV. 

CORRECTIONS  TO  COORDINATES 49 

Dip  of  horizon.  Refraction.  Semi- diameter.  Parallax. 
Diurnal  aberration. 

s 

CHAPTER  V. 
ROUGH  DETERMINATIONS rg 

Latitude  from  meridian  altitude.  Time  and  azimuth  from 
single  altitude.  Meridian  transits  for  time.  Orientation  and 
latitude  by  Polaris. 

xi 


xii  TABLE  OF  CONTENTS. 

CHAPTER  VI. 

PACK 

APPROXIMATE  DETERMINATIONS 79 

Circum-meridian  altitudes  for  latitude.  Time  from  single 
altitude.  Azimuth  observations  at  elongation.  Time  and 
azimuth  from  two  stars. 

CHAPTER  VII. 
INSTRUMENTS 99 

The  spirit-level.  Value  of  half  a  level  division.  Theory  of 
the  theodolite.  Repetition  of  angles.  The  sextant.  Chro- 
nometers. 

CHAPTER  VIII. 

ACCURATE  DETERMINATIONS.  , 141 

Time  by  equal  altitudes.  Precise  azimuth  with  theodolite. 
Zenith-telescope  latitudes. 

CHAPTER  IX. 

THE  TRANSIT  INSTRUMENT 168 

Preliminary  adjustments.  Theory  of  the  transit.  Ordinary 
method  for  time  determinations.  Personal  equation.  Methods 
and  accuracy  of  observation.  Time  determination  with 
reversal  on  each  star.  Azimuth  of  terrestrial  mark. 

BIBLIOGRAPHY 196 

INTRODUCTION  TO  TABLES 197 

DIFFERENTIAL  COEFFICIENTS 207 

TABLES  . .  .208 


FIELD  ASTRONOMY;;;,;  -, ; 

i 

CHAPTER  I. 

INTRODUCTORY. 

i.  Spherical  Trigonometry. — Any  three  points  on  the 
surface  of  a  sphere  determine  a  spherical  triangle,  whose 
sides  are  the  arcs  of  great  circles  joining  these  points, 
and  whose  angles  are  the  spherical  angles  included  be- 
tween these  arcs;  e.g.,  on  the  surface  of  the  earth, 
assumed  to  be  spherical  in  shape,  the  north  pole,  the  city 
of  St.  Louis,  and  the  borough  of  Greenwich,  England,  are 
three  points  making  a  spherical  triangle,  two  of  whose 
sides  are  the  arcs  of  meridians  joining  St.  Louis  and  Green- 
wich to  the  pole ;  the  third  side  being  the  arc  of  a  great 
circle  connecting  St.  Louis  and  Greenwich,  and  measur- 
ing by  its  length  the  distance  of  one  place  from  the  other. 
The  spherical  angle  at  the  pole  between  the  two  meridians 
is  the  longitude  of  St.  Louis,  while  the  angle  at  St.  Louis 
between  its  meridian  and  the  third  side  of  the  triangle 
represents  the  direction  of  Greenwich  from  St.  Louis,  a 
certain  number  of  degrees  east  of  north.  The  particular 
number  of  degrees  in  this  angle  is  to  be  found  by  solving 


2  FIELD  ASTRONOMY. 

the  triangle,  i.e.,  determining  the  magnitude  of  its  un- 
known parts  by  means  of  the  known  parts,  and  in  this 
case  we  may  suppose  these  known  parts  to  be  the  differ- 
ence of  longitude  between  the  two  places,  and  the  distance 
of  each  place  from  the  north  pole,  i.e.,  the  complement 
of  its  latitude. 

The  foi-mulae  required  for  the-  solution  of  a  spherical 
"triangle  afe'  best  derived  by  the  methods  of  analytical 
geometry,  and  in  Fig.  i  we  assume  a  spherical  triangle, 


FIG.  i. 

ABC,  situated  on  the  surface  of  a  sphere  whose  centre  is 
at  0,  and  we  adopt  0  as  the  origin  of  a  system  of  rect- 
angular coordinates,  in  which  the  axis  OX  passes  through 
the  vertex,  A,  of  the  triangle,  OY  lies  in  the  plane  AOB, 
and  OZ  is  perpendicular  to  that  plane.  From  the  vertex 
C  let  fall  upon  the  plane  OA  B  the  perpendicular  CP,  and 
from  P  draw  PS  perpendicular  to  OX  and  join  the  points 
C,  5,  thus  obtaining  the  right-angled  plane  triangle  CPS. 


INTRODUCTORY.  3 

The  lines  05,  SP,  PC  are  respectively  the  x,  y,  and  z 
coordinates  of  the  point  C,  and  OC,  which  we  shall  repre- 
sent by  the  symbol  r,  is  the  radius  of  the  sphere. 

It  is  evident  from  the  construction  that  the  points 
0,  S,  A,  and  C  all  lie  in  the  same  plane.  Also,  0,  5,  A, 
B,  and  P  lie  in  another  plane,  and  the  angle  between  these 
two  planes  is  measured  both  by  the  spherical  angle  BAG 
and  by  the  plane  angle  CSP,  and  these  angles  must  there- 
fore be  equal  each  to  the  other.  We  may  now  express 
the  coordinates  of  the  point  C  in  terms  of  the  sides,  a,  b,  c, 
and  angles,  Ay  B,  C,  of  the  spherical  triangle  as  follows: 

OS  =  x  =  r  cos  b, 

SP=y  =  r  sin  b  cos  A,  (i) 

PC  =  z  =  r  sin  b  sin  A. 

If  the  axis  of  x,  instead  of  passing  through  A,  had  been 
made  to  pass  through  B,  as  is  shown  by  the  broken  line 
OX' ,  the  axis  of  Z  remaining  unchanged,  we  should  have 
had  for  the  coordinates  of  C  in  this  system, 

xf  =     r  cos  a, 

y  =  —  r  sin  a  cos  B,  (2) 

z'  =     r  sin  a  sin  B. 

For  the  sake  of  simplicity  each  angle  of  the  triangle  ABC 
has  been  made  less  than  90°,  and  the  point  P,  therefore, 
falls  between  the  axes  OX,  OX' ,  thus  giving  y  and  yr 
opposite  signs,  as  shown  above. 

It  is  evident  from  the  figure  that  the  relations  between 
#»  x> '  j  y>  y'  >  are  those  furnished  by  the  formulae  for  the 


4  FIELD  ASTRONOMY. 

transformation  of  coordinates  in  a  plane,  when  the  origin 
remains  unchanged  and  the  axes  are  revolved  through  an 
angle,  which  in  this  case  is  measured  by  the  side  c  of  the 
spherical  triangle.  We  have,  therefore, 

*'=z, 

yf  —y  cos  c  —  x  sin  c,  (3) 

x*  =  y  sin  c  +  x  cos  c ; 

and  introducing  into  these  equations  the  values  of  the 
coordinates  above  determined  and  dividing  through  by 
r,  we  obtain  the  following  relations  among  the  sides  and 
angles  of  the  triangle : 

sin  a  sin  B  =  sin  b  sin  A , 

sin  a  cos  B  =  cos  b  sin  c  —  sin  b  cos  c  cos  A,  (4) 

cos  a  =  cos  b  cos  c  +  sin  b  sin  c  cos  A . 

These  are  the  fundamental  equations  of  spherical 
trigonometry  and  hold  true  not  only  for  the  particular 
triangle  for  which  they  have  been  derived,  but  for  every 
spherical  triangle,  whatever  its  shape  or  size. 

2.  Numerical  Applications  of  Equations  4. — We  proceed 
to  apply  these  equations  to  the  logarithmic  solution  of 
the  triangle  above  described,  premising  that  in  this  solu- 
tion the  signs  of  all  the  trigonometric  functions  must  be 
carefully  heeded,  since  upon  them  depend  the  quadrants, 
first,  second,  third,  or  fourth,  in  which  the  unknown  parts 
of  the  triangle  are  to  be  found.  In  this  connection  we 
shall  reserve  the  signs  +  and  —  for  natural  numbers  and 
place  after  a  logarithm  the  letter  n  whenever  the  number 
corresponding  to  the  logarithm  is  negative.  The  student 


INTRODUCTORY. 


5 


should  accustom  himself  to  this  practice,  since  it  is  the 
one  in  general  use. 

The  assumed  data  of  the  problem  are : 

Angular  distance,  Greenwich  to  Pole.  ...    b  =3 8°. 5, 

Angular  distance,  St.  Louis  to  Pole c  =  5i°.4; 

Spherical  angle  at  North  Pole A  =go°.4. 

and  these  data  we  treat  as  follows : 


Logarithms, 
sin  A  =0.000 
sin   6=9.794 
cos  A  =7.8447* 
cos  £=9.795 
sm  b  cos  A  =7.638?* 
cos  6=9.894 
sin  £=9.893 


SOLUTION. 

Numbers. 

cos  b  sin  c    =+0.613 
sin  b  cos  c  cos  A  =  —0.003 

cos  b  cos  c=  +0.489 
sin  b  sin  c  cos  A  =  —0.003 

log  cos  a  =      9.686 


Logarithms. 
sin  a  sin  B  —  9.794 
9.852 
sin  a  cosB  =9.789 


sin  0  =  9.942 
a  =  6i°.o* 


In  the  solution  printed  above,  the  student  should 
examine  the  orderly  manner  of  the  arrangement.  Each 
number  is  labelled  to  show  what  it  is,  and  from  these 
labels  we  see  that  the  first  column  contains  the  logarithms 
of  the  several  trigonometric  functions  that  appear  in  the 
second  members  of  Equations  4.  The  second  column 
contains  natural  numbers  representing  the  values  of  the 
several  terms  contained  in  these  second  members.  These 
are  obtained  by  adding  the  proper  logarithms  shown  in 
the  first  column,  and  looking  out  the  corresponding  num- 
bers in  the  tables.  An  expert  computer  will  do  this  work 
"in  his  head"  without  writing  down  a  figure  that  is  not 
shown  in  the  printed  solution. 

At  the  bottom  of  the  second  column  is  given  log  cos  a, 
obtained  by  looking  out  the  logarithm  of  the  sum  of  the 
two  numbers  that  stand  just  above  it.  This  sum  being 
positive  shows  that  the  side  a  lies  in  either  the  first  or 


6  FIELD  ASTRONOMY. 

fourth  quadrant,  but  it  alone  cannot  decide  between  these 
two  possibilities.  We  must  now  have  recourse  to  the 
third  column,  which  gives  the  logarithms  of  the  products, 
sin  a  sin  B  and  sin  a  cos  B,  as  derived  from  the  first  and 
second  columns,  and  indicates  that  these  products  are 
positive  quantities,  since  no  n  is  appended  to  either  of  the 
logarithms.  The  products  being  positive,  the  factors 
sin  a,  sin  B,  and  cos  B  must  all  have  like  signs,  and  assum- 
ing, temporarily,  that  sin  a  is  a  positive  quantity  we 
find  that  B  must  lie  in  the  first  quadrant,  since  sin  B  and 
cos  B  are  positive  numbers.  To  obtain  its  numerical 
value  we  divide  sin  a  sin  B  by  sin  a  cos  B  (subtract 
mentally  the  corresponding  logarithms)  and  find,  as  the 
result  of  the  division,  log  tan  B  =0.00 5.  This  furnishes 
the  value  of  B  given  in  the  solution,  and  fixes  as  the 
direction  of  Greenwich  from  St.  Louis,  N.  4 5°. 3  E. 

Now,  looking  up  in  the  logarithmic  tables  the  value 
of  sin  B  (log  sin  B  =9.852),  and  dividing  it  out  from 
sin  a  sin  B,  we  obtain  the  value,  9.942,  given  in  the  solu- 
tion for  sin  a.  This  number  might  equally  have  been 
obtained  by  looking  up  in  the  tables  the  values  of  cos  B 
(  =  9.847)  and  dividing  it  out  from  sin  a  cos  B,  and  with 
reference  to  this  double  possibility  the  label  for  the  line 
between  sin  a  sin  B  and  sin  a  cos  B  is  omitted,  it  being 
understood  that  either  sin  B  or  cos  B,  whichever  is  the 
greater  of  the  two,  will  be  entered  here  and  used  in  the 
proper  manner  to  obtain  sin  a.  The  value  of  log  cos  a 
is  given  in  the  middle  column,  and  both  sin  a  and  cos  a 
being  positive  numbers,  a  is  to  be  taken  in  the  first  quad- 
rant. The  agreement  between  the  numerical  values  of  a 


INTRODUCTORY.  7 

furnished  by  the  sine  and  cosine,  is  a  check  upon  the  accu- 
racy of  the  computation,  and  an  asterisk  or  check-mark 
is  placed  after  the  value  of  a  to  show  that  this  check  has 
been  applied  and  found  satisfactory. 

In  determining  the  quadrant  of  J5,  sin  a  was  assumed 
to  be  a  positive  number.  It  might  equally  well  have  been 
assumed  a  negative  number,  which  would  have  made 
sin  B  and  cos  B  both  negative,  and  would  have  furnished 
as  the  solution  of  the  triangle  B  =  225°. 6,  a^2gg°.i. 
This  is  also  a  correct  result,  for  if  we  travel  from  St.  Louis 
in  the  direction  N.  2 2 5°. 6  E.,  over  an  arc  of  a  great  circle 
299°. i  long,  we  shall  find  Greenwich  at  the  end  of  it. 
The  first  solution  represents  the  least  distance,  the  second 
solution  the  greatest  distance,  on  the  surface  of  the  sphere, 
between  the  two  points,  and  as  a  matter  of  convenience 
it  is  customary  to  use  the  first  solution  and  to  assume  that 
sin  a  is  a  positive  number. 

3.  Analytical  Applications  of  Equations  4. — Equations 
4  suffice  for  the  solution  of  any  triangle  in  which  there 
are  given  two  sides  and  the  included  angle,  but  they  are 
not  immediately  applicable  when  other  parts  of  the  tri- 
angle are  the  data  of  the  problem,  e.g.,  when  the  three 
sides  are  given  and  the  angles  are  required.  A  large  part 
of  spherical  trigonometry,  therefore,  consists  in  purely 
analytical  transformations  of  these  equations  into  forms 
adapted  to  different  data.  For  the  particular  case  above 
suggested,  a,  b,  and  c  given,  we  find  from  the  last  of 
Equations  4 

cos  a  —  cos  b  cos  c 
cos  A  =      — -  ,  (5) 


8  FIELD  ASTRONOMY. 

by  means  of  which  the  angle  A  may  be  computed  with 
the  given  data,  and  similar  equations  may  be  written  for 
the  other  angles. 

By  transformations  more  tedious  than  difficult,  and 
involving  the  introduction  of  two  auxiliary  quantities, 
denned  below  by  Equations  6,  we  may  change  Equation  5 
into  a  form  more  convenient  for  computation  when  all 
three  of  the  angles  are  to  be  determined  (see  any  treatise 
on  spherical  trigonometry  for  the  analytical  processes 
involved) .  As  a  result  of  these  transformations  we  have 
the  following  auxiliaries, 


.   (6) 


which  determine  the  angles  through  the  relations 


cot  \A  =  k  sin  0  —  a) 

cot  %B=k  sin  (s-b),  (7) 

cot  \C  =  k  sin  (s  —  c). 

Right-angled  Spherical  Triangles.  —  Since  Equations  4 
hold  true  for  all  spherical  triangles,  we  may  apply  them 
to  the  special  case  of  a  triangle  right-angled  at  A,  i.e.,  one 
in  which  the  angle  A  equals  90°.  We  shall  then  have 
sin  A  =  i,  cos  A=o,  and  with  these  special  values  we 
obtain  by  substitution  the  following  equations,  which 
should  be  compared  with  the  corresponding  formulae 
of  plane  trigonometry  : 


INTRODUCTORY. 

From  the  first  equation,  sin  B  =Sm 

sin  a' 

From  first  and  second  equations,  tan  B  = 


sin  c      (8) 

From  second  and  third  equations,  cos  B  =  JH1£ 

tana 

From  the  third  equation,         cos  a  =  cos  b  cos  c. 

These  equations  together  with  those  derived  in  the 
preceding  sections,  while  far  from  covering  the  whole  field 
of  spherical  trigonometry,  will  be  found  sufficient  for  the 
purposes  of  this  work. 

4.  Approximate  Formulae. — In  an  important  class  of 
cases  all  the  preceding  formulae  may  be  greatly  simplified 
for  numerical  use  as  follows :  It  is  shown,  in  treatises  on 
the  differential  calculus,  that  the  trigonometric  functions 
may  be  developed  in  series;  e.g., 

x3      x5 

sin  00=00  —  -r-\ —  etc., 

6      120 

X          2X^ 

tan*  =  x  +  —  +  —  +  etc.,  (9) 

O  3 

x*      x4 
cos  x  =  i  —  — f-  —  —  etc., 

2  24 

where  x  is  expressed  in  radians  (one  radian  =  5  7°.  3, 
=  3437'-75>  =206264".%).  When  the  angle  x  does  not 
exceed  a  few  minutes  of  arc,  x  radians  is  a  small  fraction, 
and  its  powers,  x1,  x3,  etc.,  are  still  smaller  quantities,  so 
that  in  these  series  we  may  suppress  all  terms  save  the 
first,  or  all  terms  save  the  first  two,  and  the  error  pro- 
duced by  neglecting  these  terms  of  a  higher  order,  as 


10  FIELD  ASTRONOMY. 

they  are  called,  is  approximately  measured  by  the  first 
term  thus  neglected.  For  illustration  we  assume  x  =  i°, 
and  turning  this  into  radians  find  the  results  shown  in 
the  following  short  table : 

Radians.  Arc. 

#=0.0174533+=  i° 

^=0.0001523+=  3i".42  (IQ) 

^x3  =0.0000009  +=  o".i8 

•£$x4  =0.0000000  +=  o".oo 

It  appears  from  the  value  of  £#3,  here  given,  that  if  we 
are  prepared  to  tolerate  in  our  work  an  error  of  one  part 
in  a  million  we  may,  for  an  arc  of  i°,  substitute  the  arc 
itself  in  place  of  its  sine,  in  any  formula  where  the  latter 
occurs ;  and  similarly  (from  the  value  of  |#2)  we  may  sub- 
stitute unity  in  place  of  the  cosine  of  an  arc  of  i  °,  if  we  are 
willing  to  admit  an  error  of  one  part  in  seven  thousand. 
Expressed  in  arc  these  errors  are  as  shown  in  the  table, 
o".2  and  3 1 ".4  respectively,  and  with  reference  to  these 
numbers  we  may  establish  the  approximate  relations: 
the  square  of  a  degree  equals  a  minute;  the  cube  of  a 
degree  equals  a  second ;  and  find  readily,  from  these  rela- 
tions, the  square  and  cube  of  any  small  arc,  and  thus 
decide  whether,  in  a  given  case,  these  quantities  may  or 
may  not  be  neglected.  For  example:  if  00  =  2°,  we  find 
x2  =  4f,  #3  =  8",  and  for  any  work  in  which  the  data  can 
be  depended  upon  to  the  nearest  minute  only,  we  may 
assume  sin  %  =x,  but  we  cannot  assume  cos  %  =  i  without 
sacrificing  some  of  the  accuracy  contained  in  the  data. 

It  must  be  constantly  borne  in  mind  that  the  series 
given  above  are  expressed  in  radians,  and  that  when 
applied  numerically,  x  and  its  powers  must  be  trans- 


INTRODUCTORY.  11 

formed  from  arc  into  radians  by  dividing  by  the  appro- 
priate factors  given  above;  e.g., 


The  divisor  given  above  is  numerically  equal  to  the 
reciprocal  of  the  sine  of  i",  and  in  place  of  the  preceding 
equation  it  is  customary  to  write 

x  (radians)  =x"  sin  i".  (12) 

As  these  numerical  factors  are  of  frequent  use,  we  record 
here  their  values  : 

log  sin  i"  =4.6855749  —  10, 

206264.  8  =  [5.  3144251].  (I3) 

Observe  the  peculiar  notation  of  the  last  line.  The 
brackets  indicate  that  the  number  placed  within  them  is 
a  logarithm,  and  the  equation  asserts  that  this  bracketed 
number  is  the  logarithm  of  the  first  member  of  the  equa- 
tion. This  use  of  the  brackets  is  very  common  and 
should  be  remembered. 

We  may  apply  to  the  equations  of  spherical  trigo- 
nometry the  principles  here  developed,  and  assuming  that 
the  sides,  a,  b,  c,  do  not  much  exceed  i°,  i.e.,  that  for  a 
triangle  on  the  surface  of  the  earth  the  vertices  of  the 
angles  are  not  more  than  sixty  or  seventy  miles  apart,  we 
shall  find  that  Equations  8  become 

b  c  b 


=  -      cos      =  -, 
a  a  c 


12  FIELD  ASTRONOMY. 

These  are  the  formulae  of  plane  trigonometry,  and  indicate 
that  small  spherical  triangles  may  be  treated  as  if  they 
were  plane. 

The  use  of  these  approximate  relations  is  not  limited 
to  the  solution  of  triangles,  but  they  may  be  applied  to 
the  trigonometric  functions  of  any  small  angle  wherever 
found,  and  we  shall  have  frequent  occasion  to  use  them 
in  the  following  pages. 

5.  Numerical  Computations. — Engineer  and  astronomer 
alike  should  acquire  the  art  of  rapid  and  correct  compu- 
tation, and  as  a  means  to  that  end  there  will  be  found 
on  subsequent  pages  examples  of  numerical  work  which 
should  be  studied  with  reference  to  their  arrangement  and 
the  order  in  which  the  several  processes  were  executed. 
Often  the  order  in  which  this  work  was  done  is  not  the 
order  in  which  the  numbers  appear  upon  the  printed 
page,  although  their  arrangement  upon  the  page  always 
follows  exactly  the  original  computation,  and  in  no  case 
is  to  be  regarded  as  a  mere  summary  of  results,  picked 
out  and  rearranged  after  the  actual  ciphering  had  been 
performed.  For  illustration  we  revert  to  the  example 
of  §  2,  and  note  that  sin  A  is  the  first  number  written  in 
the  solution  and  sin  b  stands  second.  But  the  second 
number  actually  written  down  in  the  computation  was 
cos  A,  instead  of  sin  b,  for,  having  found  the  place  in 
which  to  look  up  sin  A,  it  is  more  convenient  and  more 
economical  to  look  up  cos  A  at  once,  while  the  tables  are 
open  at  the  right  place,  rather  than  to  turn  away  for 
something  else  and  then  have  again  to  find  the  page 
and  place  corresponding  to  the  angle  A.  Having  finished 


INTRODUCTORY.  13 

with  the  required  functions  of  A ,  sin  b  was  next  looked 
out  and  was  followed  by  cos  b,  although  this  required 
the  computer  to  skip  two  intervening  lines  of  the  com- 
putation and,  temporarily,  to  leave  them  blank. 

The  general  principle  here  observed  is:  When  a  table 
is  open  at  a  given  place,  look  up,  before  leaving  it,  all 
that  is  to  be  taken  from  that  place.  In  order  to  do  this 
it  is  necessary  to  block  out  the  computation  in  advance, 
and  this  was  done  in  the  case  under  consideration,  every 
label,  from  the  initial  sin  A  of  the  first  column  to  the 
concluding  a  of  the  last  column,  being  written  in  its 
appropriate  place  before  a  number  was  set  down  or  the 
logarithmic  table  opened.  The  form  of  computation 
thus  prearranged  is  called  a  schedule,  and  it  is  to  be 
strongly  urged  upon  the  student  as  a  measure  of  econ- 
omy and  good  practice,  that  he  should  draft,  at  the 
beginning  of  each  computation,  a  complete  schedule,  in 
which  every  number  to  be  employed  shall  be  assigned 
the  place  most  convenient  for  its  use.  In  general  the 
beginner  will  not  be  able  to  do  this  without  assistance 
from  an  instructor,  or  from  models  suitably  chosen,  and 
for  the  purposes  of  the  present  work  the  numerous  exam- 
ples contained  in  the  text  may  be  taken  as  such  models. 

Some  cardinal  points  in  the  arrangement  of  a  good 
schedule  are  as  follows: 

(A)  Make  it  short  but  complete.  Do  as  much  of  the 
work  "in  your  head"  as  can  be  done  without  unduly 
burdening  the  mind,  and  write  upon  paper  only  the 
things  that  are  necessary.  But  all  things  that  are  to  be 
written  should  have  places  assigned  them  in  the  schedule. 


14  FIELD  ASTRONOMY. 

No  side  computations,  upon  another  piece  of  paper, 
should  be  allowed,  and  the  entire  work  should  be  so 
arranged  and  labelled  that  a  stranger  can  follow  it  and 
tell  what  has  been  done. 

(B)  When  the  same  quantity  is  to  be  used  several 
times  in  a  computation  (sin  b  appears  as  a  factor  in  three 
different  terms  of  the  preceding  example)  the  schedule 
should  be  so  arranged  that  the  number  need  be  written 
only  once,  e.g.,  since  sin  b  is  to  be  multiplied  by  both 
sin  A  and  cos  A   it  is  placed  between   these   numbers 
in  the  schedule,  and  for  a  similar  reason  the  product 
sin  b  cos  A  is  placed  between  cos  c  and  sin  c.     In  adding 
the    logarithms   to   form   the   product   sin  b  sin  c  cos  A , 
cover  cos  b  with  a  pencil  or  penholder  and  the  addition 
will  be  as  easily  made  as  if  the  intervening  number  were 
not  present. 

(C)  Frequently,  several  similar  computations  are  to 
be  made  with  slightly  different  data,  e.g.,  it  may  be  re- 
quired to  find  the  direction  and  distance  of  half  a  dozen 
American    cities    from    Greenwich.     A    single    schedule 
should  then  be  prepared  and  the  several  computations 
should  be  carried  on  simultaneously,  in  parallel  columns, 
all  placed  opposite  the   same   schedule;  e.g.,   look  out 
sin  A  and  cos  A  for  all  six  places  before  proceeding  to 
find  sin  b  for  any  of  them,  etc.     In  this  particular  case 
sin  b  and  cos  6,  depending  on  the  latitude  of  Greenwich, 
are  the  same  for  all  the  solutions,  and  instead  of  writing 
their  values  in  each  column,  they  should  be  written  upon 
the  edge  of  a  slip  of  paper  and  moved  along  from  column 
to  column  as  needed.     As  a  memorandum  for  future 


INTRODUCTORY.  15 

reference  they  should  also  be  written  in  one  column  of 
the  computation.  Practise  this  device  whenever  the 
same  number  is  to  be  used  in  several  different  places. 
See  §§36  and  40  for  examples  of  two  computations  de- 
pending upon  a  single  schedule. 

6.  The  Trigonometric  Functions. — There  is  opened  to 
the  inexperienced  computer  an  abundant  opportunity  for 
error  in  looking  out  from  the  tables  the  trigonometric 
functions  of  angles  not  lying  in  the  first  quadrant.  The 
best  mode  of  guarding  against  such  errors  is  the  acqui- 
sition of  fixed  habits  of  procedure,  so  that  the  same  thing 
shall  always  be  done  in  the  same  way,  and  to  this  end 
the  following  simple  rules  may  be  adopted  : 

(1)  For  any  odd-numbered  quadrant,  first,  third,  etc. 
Reduce  the  given  angle  to  the  first  quadrant  by  casting 
out  the  nines  from  its  tens  and  hundreds  of  degrees  (add 
these  digits  together  and  repeat  the  addition  until  the 
sum  is  reduced  to  a  single  digit,  less  than  nine),  and  look 
up  the  required  function  of  the  reduced  arc. 

(2)  For  any  even-numbered  quadrant,  second,  fourth, 
etc.     Reduce  the  angle  to  the  first  quadrant,  as  above, 
and  look  out  the  function  complementary  to  the  one 
given.     The  algebraic  sign  of  the  function  is,  of  course, 
in  all  cases  determined  by  the  quadrant  in  which  the 
original  angle  falls. 

See  the  following  applications  of  these  rules: 

Quadrant.  Required.                      Equivalent.  Process. 

2d,    Even  005144°   29'  =  —  sin  54°   29'  1+4=5 

3d,    Odd  tan  264°  33'  =  +  tan  84°  33'  2+6=8 

4th,Even  sin  316°  51'  =  -00546°  51'  3  +  I=4 

5th,  Odd  cot  414°   1 8'  =  +  cot  54°   18'  4  +  1  =5 

6th,  Even  tan  499°  49'  =  —cot  49°  49'  Reject  the  9 

etc.  etc.                            etc.  etc. 


16  FIELD  ASTRONOMY 

We  may  readily  formulate  a  corresponding  rule  for 
the  converse  process,  of  passing  from  the  function  to 
the  angle,  as  follows  : 

(1)  When  the  arc  lies  in  an  odd  quadrant.     Look  out, 
in  the  first  quadrant,  the  angle  that  corresponds  to  the 
given  function  and  add  to  it  the  required  even  multiple 
of  90°,  i.e.,  o°  or  180°. 

(2)  When  the  arc  lies  in  an  even  quadrant.     Change 
the  name  of  the  function  (for  cos  read  sin,  for  tan  read 
cot,  etc.).     Look  out,  in  the  first  quadrant,   the  corre- 
sponding angle  and  add  to  it  the  required  odd  multiple 
of  90°,  i.e.,  90°  or  270°. 

See  the  following  examples,  in  which  we  represent 
the  required  angle  by  z  and  suppose  that  there  is  given 
the  numerical  value  of  its  tangent,  e.g.,  log  tan  2  =  9.654. 
The  process  of  looking  out  in  the  several  quadrants  the 
angle  corresponding  to  this  tangent  is  as  follows  : 


Quadrant. 

Use. 

Angle. 

Add. 

Result. 

2d,  Even 

cot 

65°  45' 

90° 

155°  45' 

3d,  Odd 

tan 

24°   15' 

1  80° 

204°   15' 

4th,  Even 

cot 

65°  45' 

270° 

335°  45' 

etc. 

etc. 

etc. 

etc. 

etc. 

The  degrees,  minutes,  and  seconds  of  the  required  angle 
should  be  obtained  from  the  table,  the  multiple  of  90° 
added  to  them,  and  the  final  result  written  down  in  its 
proper  place,  without  writing  the  intermediate  steps. 

7.  Determination  of  Angles.  —  In  this  connection  the 
student  will  do  well  to  examine  the  beginning  of  a  table 
of  logarithmic  trigonometric  functions  and  observe  how 
difficult  it  is  to  interpolate  accurately  the  value  of 
log  sin  z  or  log  tan  z,  corresponding  to  a  small  angle,  e.g., 


INTRODUCTORY.  17 

£  =  o°  33'  17".  The  difficulty  comes  from  the  rapid  varia- 
tion of  the  function,  large  and  changing  tabular  differ- 
ences. On  the  other  hand,  log  cos  z  changes  slowly  and 
may  be  readily  and  accurately  interpolated.  If  we  take 
the  converse  case  and  suppose  the  logarithmic  function 
to  be  given  and  the  corresponding  angle  required,  we 
shall  obtain  the  opposite  result.  The  angle  will  be  accu- 
rately determined  by  the  sine  or  tangent  and  very  poorly 
determined  by  the  cosine,  e.g.,  log  cos  o°  33'  17"  =  9.99998 
and  every  angle  between  o°  29'  and  o°  36'  has  this 
cosine,  thus  leaving  a  possible  error  of  several  minutes 
in  the  value  of  the  angle  determined  from  this  function, 
while  the  log  sin,  if  correctly  given  to  five  decimal  places, 
will  determine  the  same  angle  within  a  small  fraction  of 
a  second. 

In  the  interest  of  precision  an  angle  should  always  be 
determined  from  a  function  that  changes  rapidly  (large 
tabular  differences),  while  a  quantity  that  is  to  be  found 
from  a  given  angle  is  best  determined  through  a  function 
that  changes  slowly.  In  the  example  of  §  2,  sin  a  might 
have  been  determined  through  sin  B  or  cos  B,  and  the 
former  was  used  for  this  purpose  because  it  varied  the 
more  slowly.  In  cases  of  this  kind,  and  they  are  very 
common,  use  the  function  that  stands  on  the  right-hand 
side  of  the  page,  in  the  tables,  and  subtract  it  from  the 
larger  of  the  two  numbers,  sin  a  sin  B  or  sin  a  cos  B,  and 
it  will  be  then  unnecessary  to  consider  whether  it  is  sine 
or  cosine  that  is  employed. 

The  angle  a,  in  this  example,  was  determined  through 
its  tangent  (log  tan  a  =  log  sin  a  — log  cos  a),  since  the 


18  FIELD  ASTRONOMY. 

tangent  always  varies  more  rapidly  than  either  sine  or 
cosine  and  should  generally  be  preferred  for  this  purpose. 
After  obtaining  a,  its  sine  and  cosine  were  looked  out 
from  the  tables  and  compared  with  the  numbers  obtained 
in  the  solution,  for  the  sake  of  the  "check"  thus  fur- 
nished upon  the  accuracy  of  the  numerical  work.  In 
subsequent  pages  other  checks  will  be  shown,  and  these 
should  be  applied  to  test  the  accuracy  of  numerical  work 
whenever  they  are  available.  The  mental  strain  accom- 
panying a  long  computation  is,  under  the  best  of  circum- 
stances, considerable,  and  a  check  properly  satisfied 
serves  to  relieve  this  tension  and  facilitate  the  subse- 
quent work. 

8.  Accuracy  of  Logarithmic  Computation. — The  exam- 
ple of  §  2  was  solved  with  logarithms  extending  only  to 
three  places  of  decimals,  and  corresponding  to  this  use  of 
a  three-place  table  the  results  are  given  to  the  nearest 
tenth  of  a  degree.  If  it  were  required  to  obtain  results 
correct  to  the  nearest  minute  or  nearest  second,  a  greater 
number  of  decimals  must  be  employed  (four-,  five-,  or  six- 
place  tables).  The  labor  of  using  these  tables  increases 
very  rapidly  as  the  number  of  decimals  is  increased,  and 
a  compromise  is  always  to  be  made  between  extra  labor 
on  the  one  hand  and  limited  accuracy  on  the  other. 

As  the  choice  of  a  proper  number  of  decimal  places 
is  usually  an  embarrassing  one  for  the  beginner,  there  is 
given  below  for  his  guidance  a  formula  intended  to  repre- 
sent, at  least  approximately,  the  limit  of  error  to  be  ex- 
pected in  the  results  of  computation  on  account  of  the 
inherent  imperfections  of  logarithms  (neglected  deci- 


INTRODUCTORY.  19 

mals,  etc.").  The  actual  error  may  fall  considerably 
short  of  this  limit  or  may  overstep  it  a  little.  It  is 
evident  that  the  limit  will  be  greater  for  a  long  compu- 
tation than  for  a  short  one,  and  if  we  measure  the  length 
of  a  computation  by  the  number,  n,  of  logarithms  that 
enter  into  it  and  represent  by  m  the  number  of  decimal 
places  to  which  these  logarithms  are  carried,  there  may 
be  derived  from  the  theory  of  probabilities  the  following 
expression,  in  minutes  of  arc,  for  the  limit  of  probable 
error: 

Limit  =  2800'  .  Vn  .  io~m. 

Applying  this  formula  to  the  example  of  §  2  we  may 
put  n  =  i6,  m  =  3,  and  find  10'  as  the  limit  of  unavoid- 
able error;  corresponding  well  with  the  one-tenth  of  a 
degree  to  which  the  results  were  carried.  If  the  data 
were  given  to  the  nearest  minute  and  it  were  required 
to  preserve  this  degree  of  accuracy  in  the  results,  we 
should  write, 


and  solving,  find  ^  =  4.0,  i.e.,  a  four-place  table  is  re- 
quired for  this  purpose. 

Let  the  student  verify  by  means  of  the  above  equa- 
tions the  following  precepts: 

To  obtain  Use 

Tenths  of  degrees  Three-place  tables. 

Minutes  Four-place  tables. 

Seconds  Five-  or  six-place  tables. 

Tenths  of  seconds  Seven-place  tables. 


•30  FIELD  ASTRONOMY. 

If  the  results  are  to  be  expressed  in  linear  instead  of 
angular  measure,  the  limit  of  error  must  be  represented 
as  a  fractional  part  of  the  quantity,  x,  that  is  to  be 
determined,  and  corresponding  to  this  case  we  have 

Limit  =  o .  8  # .  Vn .  io~M. 

Corollary.  Do  not  attempt  to  obtain  from  a  table 
more  than  it  is  capable  of  furnishing;  e.g.,  do  not  inter- 
polate hundredths  of  a  degree  in  the  example  of  §  2, 
and  in  connection  with  linear  quantities  do  not,  as  a  rule, 
interpolate  more  than  three  significant  figures  from  a 
three-place  table,  four  from  a  four-place  table,  etc. 

9.  Logarithmic  Tables. — There  exists  a  great  variety 
of  logarithmic  tables  of  different  degrees  of  accuracy, 
from  three  to  ten  places  of  decimals,  and  having  deter- 
mined the  number  of  decimal  places  required  in  a  given 
computation,  the  choice  among  the  corresponding  tables 
is  largely  a  matter  of  personal  taste.  The  beginner, 
however,  will  do  well  to  observe  the  following  rules  foj 
distinguishing  good  tables  from  bad  ones: 

(A)  Wherever    the    tabular    differences    exceed    10, 
a   good   table    should   furnish   proportional   parts,    PP, 
in  the  margin  of  each  page,  so  that  the  logarithms  may 
be  interpolated  ''in  the  head." 

(B)  The  tables  should  be  accompanied  by  tables  of 
addition  and  subtraction  logarithms.     For  an  explana- 
tion of  these,  their  purpose  and  use,  the  student  is  re- 
ferred to  the  tables  themselves,  but  we  note  here  that 
by  their  aid  the  example  of  §  2  might  have  been  much 


INTRODUCTORY.  21 

more  conveniently  solved,  as  is  illustrated  in  a  similar 
problem  in  §  15. 

The  most  generally  useful  tables  are  those  of  five 
decimal  places,  but  computers  find  it  to  their  advantage 
to  have  and  use  at  least  one  table  of  each  kind,  from  three 
to  six  or  seven  places.  In  the  examples  solved  in  the 
present  work  the  following  tables  have  been  used : 

Three-place,  Johnson.     New  York. 

Four-place,  Slichter.      New  York.     Gauss.    Berlin. 

Five-place,  Albrecht.     Berlin. 

Six-place,  Albrecht's    Bremiker.      Berlin. 

As  a  very  useful  supplement  to  the  logarithmic  tables 
a  slide-rule  and  the  extended  multiplication  tables  of 
Crelle  and  Zimmermann  are  highly  esteemed  and  where 
extensive  computations  are  to  be  made  much  advantage 
may  be  derived  through  the  use  of  computing  machines, 
of  which  several  types  are  to  be  had. 


CHAPTER   II. 

COORDINATES. 

10.  Fundamental  Concepts.  —  For  most  purposes  of 
practical  astronomy  the  stars  may  be  considered  as 
attached  to  the  sky,  i.e.,  to  the  blue  vault  of  the  heavens, 
which  is  technically  called  the  celestial  sphere,  and  is  re- 
garded as  of  indefinitely  great  radius  but  having  the 
earth  at  its  centre,  so  that  a  plane  passing  through  any 
terrestrial  point  intersects  this  sphere  in  a  great  circle, 
and  parallel  planes  passing  through  any  two  terrestrial 
points  intersect  the  sphere  in  the  same  great  circle. 

If  the  axis  about  which  the  earth  rotates  be  produced 
in  each  direction,  it  will  intersect  the  celestial  sphere  in 
two  points,  called  respectively  the  north  and  south  poles. 
If  a  plumb-line  be  suspended  at  any  place,  P,  on  the 
earth's  surface,  and  be  produced  in  both  directions,  it 
will  intersect  the  celestial  sphere,  above  and  below,  in 
the  zenith  and  nadir  of  the  place.  The  direction  thus 
determined  by  the  plumb-line  is  called  the  vertical  of  the 
place. 

The  figure  (shape)  of  the  earth  is  such  that  the  vertical 
of  any  place,  when  produced  downward,  intersects  the 

rotation  axis,  and  a  plane  may  therefore  be  passed  through 

22 


COORDINATES.  23 

this  axis  and  the  vertical.  This  plane,  by  its  intersection 
with  the  celestial  sphere,  produces  a  great  circle  which 
passes  through  the  poles,  the  zenith  and  nadir,  and  is 
called  the  meridian  of  the  place,  P. 

A  plane  passed  through  P  perpendicular  to  the  direc- 
tion of  the  vertical  produces  by  its  intersection  with  the 
sphere  the  horizon  of  P.  Any  plane  passing  through  the 
vertical  is  called  a  vertical  plane  and  produces  by  its  inter- 
section with  the  sphere,  a  vertical  circle.  That  vertical 
circle  whose  plane  is  perpendicular  to  the  meridian  is 
called  the  prime  vertical. 

With  exception  of  the  poles,  all  of  the  terms  above 
denned  depend  upon  the  direction  of  the  vertical,  and 
as  this  direction  varies  from  place  to  place  upon  the 
earth's  surface  each  such  place  has  its  own  meridian, 
horizon,  zenith,  etc.,  while  the  poles  of  the  celestial 
sphere  are  the  same  for  all  places. 

A  plane  passed  through  the  centre  of  the  earth  per- 
pendicular to  the  rotation  axis  produces  by  its  inter- 
section with  the  earth's  surface  the  terrestrial  equator, 
and  by  its  intersection  with  the  celestial  sphere  it  pro- 
duces the  celestial  equator. 

Owing  to  the  motion  of  the  earth  in  its  orbit  we  see 
anything  within  the  orbit  from  different  points  of  view 
at  different  seasons  of  the  year,  and  by  the  earth's  motion 
the  sun  is  thus  made  to  describe  an  apparent  path  among 
the  stars,  making  the  complete  circuit  of  the  sky  in  a 
year.  This  path  is  a  great  circle  intersecting  the  celestial 
equator  in  two  points  diametrically  opposite  to  each 
other,  and  that  one  of  these  points  through  which  the 


24  FIELD  ASTRONOMY. 

sun  passes  on  or  about  March  22  of  each  year,  is  called 
the  vernal  equinox. 

ii.  Systems  of  Coordinates.  —  Most  of  the  problems 
of  practical  astronomy  require  us  to  deal  with  the  appar- 
ent positions  and  motions  of  the  heavenly  bodies  as 
seen  projected  against  the  sky,  and  for  this  purpose 
there  are  employed  several  systems  of  coordinates  based 
upon  the  concepts  above  denned,  and  three  of  these 
systems  we  proceed  to  consider.  These  are  all  systems 
of  polar  coordinates  having  the  following  characteristics 
in  common: 

1 i )  The  origin  of  each  system  is  at  the  centre  of  the 
celestial  sphere. 

(2)  Each  system  has  a  fundamental  plane  and  con- 
sists of  an  angle  measured   in  the  fundamental    plane; 
an  angle    measured    perpendicular  to  the  fundamental 
plane;  and  a  radius  vector.     The  first  of  these  angles 
is  frequently  called  the  horizontal  coordinate,  and  the 
second  the  vertical  coordinate,  of  the  system.     Latitudes 
and  longitudes  on  the  earth  furnish  such  a  system  of 
coordinates.     The    longitude    of    St.    Louis    (horizontal 
coordinate)  is  measured  by  an  angle  lying  in  the  plane 
of  the  equator,  which  is  the  fundamental  plane  of  this 
system.     The  latitude  of  St.  Louis  (vertical  coordinate) 
is  measured  by  an  angle  lying  in  a  plane  perpendicular 
to  the  equator,  and  the  radius  vector  of  St.  Louis  is  its 
distance  from  the  centre  of  the  earth,  which  latter  point 
is  taken  as  the  origin  of  coordinates. 

(3)  In    each    system    the    horizontal    coordinate    is 
measured   from   a   fixed   direction   in   the   fundamental 


COORDINATES.  25 

plane,  called  the  prime  radius,  through  360°.  The  ver- 
tical coordinate  is  measured  on  each  side  of  the  funda- 
mental plane  from  o°  to  90°. 

(4)  Those    vertical    coordinates    are    called    positive 
that  lie  upon  the   same  side  of  the  fundamental  plane 
with  the  zenith  of  an  observer  in  the  northern  hemi- 
sphere of  the  earth.     Those  that  lie  upon  the  opposite 
side  of  the  fundamental  plane  are  negative. 

(5)  It  is  frequently  convenient  to  measure  a  vertical 
coordinate  from  the  positive  half  of  a  line  perpendicular 
to  the  fundamental  plane  instead  of  from  the  funda- 
mental plane  itself,  (e.g.,  in  §  2  we  take  as  the  vertical 
coordinate  of  St.  Louis  its  distance  from  the  pole  instead 
of  from  the  equator) .     In  such  cases  this  coordinate  is 
always  positive  and  is  included  between  the  limits  o° 
and  1 80°.     If  h  represent  any  vertical  coordinate  meas- 
ured in  the  manner  first  described  and  z  be  the  corre- 
sponding coordinate  measured  in  the   second  way,  we 
shall  obviously  have  the  relation,  z  =  go°  —  h. 

The  several  systems  of  astronomical  coordinates 
differ  among  themselves  in  the  following  respects : 

(a)  Different  fundamental  planes  for  the  systems. 

(b)  Different  positions  of  the  prime  radii  in  the  fun- 
damental planes. 

(c)  Different  directions  in  which  the  horizontal  co- 
ordinates increase. 

The  data  which  completely  define  each  system  of 
coordinates  are  given  in  the  following  table  together 
with  the  names  of  the  several  coordinates,  the  letters 
by  which  these  are  usually  represented,  and  the  point 


26  FIELD  ASTRONOMY. 

of  the  heavens,  called  the  pole  of  the  system,  toward 
which  the  positive  half  of  the  normal  to  the  fundamental 
plane  is  directed.  The  terms  east  and  west  are  used 
in  this  table  with  their  common  meaning,  to  indicate 
the  direction  toward  which  the  horizontal  coordinate 
increases.  The  letters  associated  with  the  several  co- 
ordinates are  conventional  symbols  that  should  be 
committed  to  memory. 

SYSTEMS  OF   COORDINATES. 


System  

I. 

II. 

III. 

Horizon 

Equator 

Prime  radius  points  toward  

Meridian 

Meridian 

Vernal  Equinox 

Horizontal    coordinates    increase 

West 

West 

East 

Zenith 

North  Pole 

North  Pole 

Name  of  horizontal  coordinate..  . 
Name  of  vertical  coordinate  

Azimuth—  A 
Altitude=^ 

Hour  angle=^ 
Declination=8 

Right  Ascension  =  a 
Declination=5 

EXERCISES. — Let  the  student  define  in  his  own  language  the  several 
quantities  above  represented  by  the  letters  A ,  t,  a,  k,  and  d. 

1 .  What  is  the  azimuth  of  the  north  pole  ? 

2.  What  do  the  hour  angle  and  altitude  of  the  zenith  respectively 
equal  ? 

3.  What  are  the  azimuths  of  the  prime  vertical? 

4.  What  are  the  declinations  of  the  points   in  which  the  horizon 
cuts  the  prime  vertical? 

5 .  Does  d  in  the  second  system  differ  in  any  way  from  d  in  the  third 
system  ? 

The  directions  of  the  prime  radius  as  above  defined 
for  systems  I  and  II,  are  ambiguous,  since  the  meridian 
cuts  the  fundamental  plane  of  each  of  those  systems 
in  two  points.  Either  of  these  points  may  be  used 
to  determine  the  direction  of  the  prime  radius,  but  in 
general  that  one  is  to  be  employed  which  lies  south  of 
the  zenith. 


COORDINATES.  27 

Let  the  student  show  the  relation  between  the  coordinates  furnished 
in  System  I  by  adopting  each  of  the  possible  positions  for  the  prime 
radius. 

12.  Uses  of  the  Three  Systems. — It  is  well  to  consider 
here,  very  briefly,  the  reasons  for  using  more  than  one 
system  of  coordinates,  and  the  relative  advantages  and 
disadvantages  of  these  systems. 

The  coordinates  of  System  I  are  well  adapted  to 
observation  with  portable  instruments,  e.g.,  an  engineer's 
transit,  since  the  horizon  is  more  easily  identified  with 
such  an  instrument  than  is  any  other  reference  plane, 
and  the  circles  of  the  instrument  may  be  made  to  read, 
directly,  altitudes  and  azimuths.  The  horizon  has  been 
defined  by  reference  to  the  direction  of  a  plumb-line, 
but  in  practice  a  spirit-level,  or  the  level  surface  of  a 
liquid  at  rest,  are  more  frequently  used  to  determine 
its  position. 

System  I  possesses  the  disadvantage  that,  through 
the  earth's  rotation  about  its  axis,  both  the  altitude  and 
azimuth  of  a  star  are  constantly  changing  in  a  compli- 
cated manner,  and  in  this  respect  System  II  possesses 
a  marked  advantage.  Since  the  normal  to  its  funda- 
mental plane  coincides  with  the  earth's  axis,  rotation 
about  this  axis  has  no  effect  upon  the  vertical  coordi- 
nates, decimations,  which  remain  unchanged,  while  the 
horizontal  coordinates,  hour  angles,  increase  uniformly 
with  the  time,  15°  per  hour,  and  are  therefore  easily 
taken  into  account  and  measured  by  means  of  a  clock. 

Suppose  a  watch  to  have  its  dial  divided  into  twenty- four  hours, 
instead  of  the  customary  twelve.  If  this  watch  be  held  with  its  dial 
parallel  to  the  plane  of  the  equator,  the  hour  hand,  in  its' motion  around 


28  FIELD  ASTRONOMY. 

the  dial,  will  follow  and  keep  up  with  the  sun  as  it  moves  across  the 
sky.  If  the  watch  be  turned  in  its  own  plane  until  the  hour  hand 
points  toward  the  sun,  the  time  indicated  upon  the  dial  by  this  hand 
will  be  approximately  the  hour  angle  of  the  sun,  and  the  zero  of  the 
dial  will  point  toward  the  meridian,  i.e.,  south. 

Let  the  student  compare  the  ideal  case  above  considered  with  the 
following  rough  rule  sometimes  given  for  determining  the  direction 
of  the  meridian  by  means  of  a  watch  with  an  ordinary,  twelve-hour, 
dial:  Hold  the  watch  with  its  dial  as  nearly  parallel  to  the  plane  of  the 
equator  as  can  be  estimated.  (See  §  13  for  the  position  of  this  plane.) 
Revolve  the  watch  in  this  plane  until  the  hour  hand  points  toward 
the  sun,  and  the  south  half  of  the  meridian  will  then  cut  the  dial  mid- 
way between  the  hour  hand  and  the  figure  XII. 

A  further  advantage  is  gained  in  the  third  system 
of  coordinates,  since  here  the  prime  radius  shares  in 
the  apparent  rotation  of  the  celestial  sphere  about  the 
earth's  axis,  and  both  the  horizontal  and  vertical  coor- 
dinates are  therefore  unaffected  by  this  motion.  In- 
struments have  been  devised  for  the  measurement  of 
the  coordinates  in  each  of  these  systems,  but  we  shall 
be  mainly  concerned  with  those  that  relate  to  the  first 
system,  and  shall  consider  System  III  as  employed 
chiefly  to  furnish  a  set  of  coordinates  independent  of  the 
earth's  rotation  and  of  the  particular  place  upon  the 
earth  at  which  the  observer  chances  to  be.  These 
features  make  it  suited  to  furnish  a  permanent  record 
of  a  star's  position  in  the  sky,  and  it  is  so  used  in  the 
American  Ephemeris  (see  §  21)  and  other  nautical  alma- 
nacs, where  there  may  be  found,  tabulated,  the  right 
ascensions  and  declinations  of  the  sun,  moon,  planets, 
and  several  hundred  of  the  brighter  stars. 

13.  Relations  between  the  Systems  of  Coordinates. — 
A  problem  of  frequent  recurrence  is  the  transformation 
of  the  coordinates  of  a  star  from  one  system  to  another; 


COORDINATES. 


indeed  most  of  the  problems  of  spherical  astronomy  are, 
analytically,  nothing  more  than  cases  of  such  trans- 
formation, and  as  an  introduction  to  these  problems 
we  shall  examine  the  relative  positions  of  the  funda- 
mental planes  and  prime  radii  of  the  several  systems. 

The  plane  of  the  equator  intersects  the  plane  of  the 
horizon  in  the  east  and  west  line,  and  the  angle  between 
the  two  planes  is  called  the  colatitude,  since  it  is  the 
complement  of  the  geographical  latitude  of  the  place 


EQUATOR 


n         \ 

/                                                                                  !   HORIZOM 

i          /\ 

\         /    \ 

\ 

X                / 

\      / 

XN           / 

FIG.  2. 


to  which  the  horizon  belongs.  The  latitude  is  commonly 
defined  as  the  angular  distance  of  any  place  from  the 
equator,  but  more  precisely  the  latitude  is  the  angle 
which  the  vertical  of  the  given  place  makes  with  the 
plane  of  the  equator.  From  Fig.  2,  which  represents 
a  meridian  section  of  the  earth  with  the  several  lines 


30  FIELD  ASTRONOMY. 

and  planes  passed  through  its  centre,  it  is  apparent 
that  the  latitude,  0,  equals  the  declination  of  the  zenith 
and  also  equals  the  altitude  of  the  pole.  The  angular 
distance  of  the  zenith  from  the  pole  is  equal  to  the  co- 
latitude,  90°  —  0. 

The  second  and  third  systems  of  coordinates  have 
the  same  fundamental  plane,  and  their  relation  to  each 
other  is  therefore  determined  by  the  angle,  0,  between 
their  prime  radii.  Since  one  of  these  prime  radii  is 
directed  toward  a  fixed  point  of  the  heavens,  while  the 
other  lies  in  a  meridian  of  the  rotating  earth,  it  is  evident 
that  the  angle  6  is  continuously  and  uniformly  variable, 
at  the  rate  of  360°  in  twenty-four  hours.  Methods  of 
determining,  for  any  instant,  the  value  of  this  angle, 
which  is  called  the  sidereal  time,  will  be  given  hereafter. 
For  the  present  we  note  that  0  may  be  regarded  as  the 
horizontal  coordinate  of  the  vernal  equinox  in  the  second 
system,  or  as  the  horizontal  coordinate  of  the  meridian 
in  the  third  system,  and  correspondingly  we  may  define 
the  sidereal  time  as  either  the  hour  angle  of  the  vernal 
equinox  or  the  right  ascension  of  the  meridian. 

14.  Transformation  of  Coordinates. — The  transforma- 
tion of  coordinates  from  the  first  to  the  second  system 
is  conveniently  made  by  means  of  the  ''astronomical 
triangle,"  i.e.,  the  spherical  triangle  formed  by  the 
zenith,  the  pole,  and  the  star  whose  coordinates  are 
to  be  transformed.  In  Fig.  3  this  triangle  is  marked 
by  the  letters  PZS,  P  indicating  the  position  of  the 
celestial  pole;  Z,  the  observer's  zenith;  and  5,  the 
apparent  place  of  the  star  as  seen  against  the  sky.  Imag- 


COORDINATES. 


31 


ine  the  triangle  projected  against  the  sky  and  the  three 
points  to  be  visible  in  their  true  positions. 

PZ  is  an  arc  of  a  great  circle  passing  through  the  pole 
and  zenith  and  must  therefore  be  a  part  of  the  observer's 
celestial  meridian,  and  in  Fig.  2  we  have  already  seen 
that  this  arc  of  the  meridian  is  equal  in  length  to  the 
complement  of  the  latitude,  90°— 0.  The  broken  line 


FIG.  3. — The  Astronomical  Triangle. 

HHf  in  the  figure,  is  an  arc  of  a  great  circle,  every  part 
of  which  is  90°  distant  from  Z.  But  the  great  circle 
90°  distant  from  the  zenith  is  the  horizon,  -and  the  arc 
HS  that  measures  the  distance  of  5  from  HH'  must  be 
the  star's  altitude,  h,  and  the  side  SZ  of  the  astronom- 
ical triangle,  being  the  complement  of  this  arc,  is  equal 
to  go°  —  h.  In  like  manner,  EE't  drawn  90°  distant 
from  P,  is  an  arc  of  the  celestial  equator;  the  a-c  SE, 
that  measures  the  distance  of  5  from  EEf ,  is  the  star's 
declination,  d,  and  the  side  PS  of  the  triangle  equals 
go°-d. 


32  FIELD  ASTRONOMY. 

The  star's  hour  angle,  i.e.,  horizontal  coordinate 
lying  in  the  equator,  is  measured  by  the  arc  EE',  and 
this  arc,  by  a  theorem  of  spherical  geometry,  is  numer- 
ically equal  to  the  spherical  angle,  t,  included  between 
EP  and  E'P,  which  is  therefore  the  star's  hour  angle. 
In  like  manner  the  spherical  angle  SZE'  is  shown  to  be 
equal  to  the  star's  azimuth,  A,  and  the  angle  SZP  of 
the  astronomical  triangle  is  equal  to  180°  —  A.  The 
third  angle  of  the  triangle,  marked  q  in  the  figure,  is 
called  the  parallactic  angle. 

To  apply  to  the  astronomical  triangle  the  fundamental 
formulae  of  spherical  trigonometry  derived  in  §  i,  we 
replace  the  general  symbols  used  in  Equations  4  by 
the  particular  values  which  they  have  in  the  astronomical 
triangle,  as  follows  : 

a  =   o°-/*  A=t 


and  introducing  these  values  into  Equations  4  we  obtain 
the  required  formulae  for  transforming  altitudes  and 
azimuths  into  declinations  and  hour  angles,  as  follows: 

cos  h  sin  A  =  +  cos  d  sin  t, 

cos  h  cos  A  =  —  cos  0  sin  d  +  sin  0  cos  d  cos  t,        (14) 
sin  h  =  +  sin  0  sin  d  +  cos  0  cos  d  cos  /. 

The  transformation  formulae  between  the  second  and 
third  systems  are  much  simpler.  In  Fig.  3  if  V  repre- 
sent the  position  of  the  vernal  equinox,  we  shall  have 
the  arc  VE',  or  the  corresponding  spherical  angle  at  P, 
equal  to  the  sidereal  time,  0,  since  the  sidereal  time  is  the 


COORDINATES.  33 

hour  angle  of  the  vernal  equinox.  Similarly  the  arc 
VE  and  its  corresponding  angle  at  P  are  equal  to  the 
right  ascension  of  the  star,  and  from  the  figure  we  then 
obtain  the  required  relations, 


0,  d  =  d  .....  (i4a) 

The  transformation  between  the  first  and  third  systems 
is  best  made  through  the  second  system;  i.e.,  by  using 
both  groups  of  formulae  14  and  i4a. 

15.  Problem  in  Transformation  of  Coordinates.  —  At 
the  sidereal  time  i3h  22™  49^3  the  altitude  and  azimuth 
of  a  star  were  measured  at  a  place  in  latitude  43°  4'  36", 
as  follows:  h  =  6  1°  19'  36",  A  =253°  9'  42".  Required 
the  right  ascension  and  declination  of  the  star. 

The  required  transformation  formulae  may  be  obtained 
from  the  astronomical  triangle  in  the  same  manner  as 
Equations  14,  and  are: 

cos  d  sin  t  =  +  cos  h  sin  A  , 
cos  d  cos  t=  +  cos  h  cos  A  sin  0  +  sin  h  cos  0, 
sin  d  =  —  cos  h  cos  A  cos  0  +  sin  h  sin  0 
a  =  6-t. 

Note  that  0,  which  occurs  only  in  the  last  equation,  is 
expressed  in  hours,  minutes,  and  seconds  of  time,  and 
that  it  is  customary  to  express  both  a  and  t  in  these  units, 
15°  =  i  h,  etc. 

A  convenient  form  for  the  numerical  operations 
involved  in  solving  these  equations  is  given  below,  and 
the  student  should  trace  it  through,  verifying  each  num- 
ber and  ascertaining  why  the  work  is  arranged  as  it  is. 


31  FIELD  ASTRONOMY. 

Compare  and  contrast  this  solution  with  the  one  con- 
tained in  §  2.  The  difference  of  arrangement  is  largely 
due  to  the  introduction  here  of  addition  and  subtraction 
logarithms.  These  are  indicated  in  the  schedule  by  the 
words  Add,  Subtract,  and  it  is  to  be  especially  borne  in 
mind  that  the  addition  indicated  by  the  word  Add 
requires  a  subtraction  logarithm  when  one  of  the  given 
terms  is  itself  a  negative  quantity,  etc.  The  schedule 
shows  the  algebraic  operation  required  by  the  formula, 
but  the  arithmetical  character  of  the  operation  is  altered 
by  the  presence  of  an  odd  number  of  negative  signs. 

SOLUTION. 


sin  A 

9  .9809772 

sin  h  cos  <£ 

9  .80676 

cos  h 

9.68108 

cos  h  cos  A  sin  <f> 

8.97740*1 

cos  A 

9  .46191*1 

Add 

0.75974 

sin'(£ 

9.83441 

cos  d  cos  t 

9-737J4 

sin  h 

9.94318 

(Seep.  6) 

9.88380 

cos  h  cos  A 

9.1429971 

cos  d  sin  t 

9  .  6620571 

COS  0 

9.86358 

t 

3  1  9°  5  5'  44" 

cos  $ 

9-85334 

sin  h  sin  <j> 

9-77759 

t  (time) 

2ihi9m428.  9 

cos  h  cos  A  cos  ^ 

9  .0065771 

6 

I3h22m49'.3 

Subtract 

o  .06797 

a 

i6h  3m  6'.  4 

sin  6 

9-84556 

8 

44°  29'  12"* 

The  declination,  d,  is  obtained  both  from  sin  d  and 
cos  d,  and  the  agreement  of  the  two  values  is  a  *  check ' 
upon  the  accuracy  of  the  work.  The  *  indicates  that 
this  check  has  been  applied. 


CHAPTER  III. 

TIME. 

1 6.  In  astronomical  practice,  time  is  measured  by 
watches  and'  clocks  that  differ  in  no  essential  respect 
from  those  in  common  use,  but  in  addition  to  the  com- 
mon system  of  time  reckoning,  astronomers  employ 
several  others,  of  which  we  shall  have  to  consider  the 
following : 

Sidereal  Time,  already  referred  to  in  §  13. 

True  Solar  Time,  which  is  frequently  called  Appar- 
ent Solar  Time. 

Mean  Solar  Time,  which  is  the  common  system  of 
every-day  life. 

These  three  systems  possess  the  following  features 
in  common:  In  each  system  that  common  phrase  ''the 
time  of  day"  means  the  hour  angle  of  a  particular  point 
in  the  heavens,  which  we  shall  call  the  zero  point  of  the 
system.  The  unit  of  time  is  called  a  day  and  is,  in  every 
case,  the  interval  between  consecutive  returns  of  the 
zero  point  to  a  given  meridian;  i.e.,  consecutive  transits' 
of  a  given  meridian  past  the  same  zero  point.  This  unit 
is  subdivided  into  aliquot  parts  called  hours,  minutes, 
and  seconds.  Each  day  begins  at  the  instant  when  the 
zero  point  is  on  the  meridian;  i.e.,  on  the  upper  half  of 

35 


36  FIELD  ASTRONOMY. 

the  meridian  (noon)  in  astronomical  practice,  on  the 
lower  half  of  the  meridian  (midnight)  in  civil  affairs.  In 
astronomical  practice  the  hours  from  the  beginning  of 
the  day  are  reckoned  consecutively,  from  o  to  24;  in 
civil  practice  from  o  to  12,  and  then  repeated  to  12  again, 
with  the  distinguishing  symbols  A.M.  and  P.M.  In  con- 
sequence of  the  different  epochs  at  which  the  day  begins, 
the  astronomical  date  in  the  A.M.  hours  is  one  day  be- 
hind the  civil  date;  e.g., 

Civil  Time May  10,    5h  A.M.     equals 

Astronomical  Time  May    9,  iyh. 

In  the  P.M.  hours  the  dates  agree. 

Since  an  hour  angle  must  be  reckoned  from  a  deter- 
minate meridian,  this  meridian  must  be  specified  in  order 
to  make  "the  time"  a  determinate  quantity,  and  this 
specification  of  the  meridian  should  be  included  in  the 
name  assigned  to  the  time;  e.g.,  Local  Time  denotes  the 
hour  angle  of  the  zero  point  reckoned  from  the  observer's 
own  (local)  meridian.  Greenwich  Time  is  the  hour  angle 
of  the  zero  point  reckoned  from  the  meridian  of  Green- 
wich. Standard  Time  is  the  hour  angle  of  the  zero  point 
reckoned  from  some  meridian  assumed  as  standard; 
e.g.,  in  the  United  States  and  Canada  the  meridians  75°, 
90°,  105°,  and  120°  west  of  Greenwich  are  called  standard, 
and  Eastern,  Central,  Mountain,  and  Pacific  Standard 
Times  are  hour  angles  reckoned  from  these  meridians. 

A  like  practice  is  followed  in  the  use  of  the  term  Noon; 
e.g.,  Washington  Noon  is  the  instant  at  which  the  zero 
point  is  in  the  act  of  crossing  the  meridian  of  Washington. 


TIME.  37 

17.  Longitude  and  Time.  —  We  have  introduced  above 
a  reference  to  the  time  at  different  meridians,  and  we  have 
now  to  note  that  since  '  '  the  time"  is  defined  as  an  hour 
angle,  it  is  evident  that  the  number  of  hours,  minutes, 
and  seconds  expressing  either  time  or  hour  angle  will 
depend  upon  the  meridian  from  which  the  latter  is  meas- 
ured. The  difference  between  the  hour  angles  reckoned 
from  two  different  meridians  will  equal  the  angle  between 
the  meridians,  i.e.,  their  difference  of  longitude,  so  that 
if  Tf  and  T"  represent  the  times  of  any  event  (whether 
sidereal,  mean  solar,  or  true  solar  time)  referred  to  two 
different  meridians  whose  difference  of  longitude  is  A, 
we  shall  have  Tr  —  T"  =  A.  It  is  customary  in  astronom- 
ical practice  to  express  differences  of  longitude  in  hours 
rather  than  in  degrees,  since  both  members  of  the  pre- 
ceding equation  should  be  given  in  terms  of  the  same 
units. 

By  transposition  of  one  term  in  the  preceding  equa- 
tion we  obtain 

r  =  r"  +  A,  ......    (16) 


and  this  extremely  simple  equation  indicates  that  any 
given  time  referred  to  the  second  meridian  may  be  re- 
duced to  the  corresponding  time  of  the  first  meridian 
by  addition  of  the  difference  of  longitude,  where  this  dif- 
ference, A,  is  to  be  counted  a  positive  quantity  when  the 
second  meridian  is  west  of  the  first.  A  very  common 
blunder  is  to  omit  this  reduction  to  the  prime  meridian 
when  interpolating  from  the  almanac  (see  §  21).  Beware 
of  it,  and  note  that  the  hour  and  minute  for  which  a  quart- 


38  FIELD  ASTRONOMY. 

tity  is  required  to  be  interpolated  are  usually  given  in 
the  time  of  some  meridian  other  than  that  of  Greenwich 
or  Washington,  for  which  the  almanac  is  constructed, 
and  must  therefore  be  reduced  to  one  of  these  standard 
meridians,  by  addition  of  the  longitude,  before  they  can 
serve  as  the  argument  for  the  tabular  quantity  sought. 

18.  The  Three  Time  Systems. — The  several  time  sys- 
tems differ  one  from  another  chiefly  in  respect  of  their 
zero  points,  and  these  we  have  now  to  consider. 

Sidereal  Time. — As  already  indicated,  the  zero  point 
of  this  system  is  the  vernal  equinox,  and  since  this  is  a 
point  of  the  heavens  whose  position  with  respect  to  the 
fixed  stars  changes  very  slowly,  it  measures  well  their 
diurnal  motion.  In  colloquial  language,  ' '  the  stars  run 
on  sidereal  time, ' '  and  this  system  is  chiefly  used  in  con- 
nection with  their  apparent  diurnal  motion. 

Solar  Time. — As  their  names  indicate,  both  True 
Solar  Time  and  Mean  Solar  Time  have  zero  points  that 
depend  upon  the  sun,  and  before  drawing  any  distinction 
between  the  two  systems  we  recall  that,  owing  to  the 
earth's  annual  motion  in  its  orbit,  the  sun's  position 
among  the  stars  changes  from  day  to  day  (we  see  it  from 
different  standpoints).  While  this  change  in  the  sun's 
position  is  not  an  altogether  uniform  one  and  takes  place 
in  a  plane  inclined  to  that  of  the  earth's  rotation  (ecliptic, 
and  equator),  its  net  result  is  that  in  each  year  the  sun 
makes  one  entire  circuit  of  the  sky,  so  that  any  given 
meridian  of  the  earth,  in  the  course  of  a  year,  makes  one 
less  transit  over  the  sun  than  over  a  star,  or  over  the 
vernal  equinox.  The  number  of  solar  days  in  a  year  is 


TIME.  39 

therefore  one  less  than  the  number  of  sidereal  days;  e.g., 
for  the  epoch  1900,  (according  to  Harkness,) 

One  (tropical)  year  =  366. 242 197  sidereal  days 

=  365.242197  solar  days.    .      (17) 

It  appears  from  this  relation  that  a  sidereal  unit  of  time 
(day,  hour,  minute)  must  be  shorter  than  the  corre- 
sponding solar  unit,  a  relation  that  we  shall  have  to  con- 
sider hereafter. 

Apparent,  or  True,  Solar  Time. — This  system  has  for 
its  zero  point  the  centre  of  the  sun,  and  the  hour  angle 
of  the  sun's  centre  at  any  moment  is,  therefore,  the  true 
solar  time.  This  system  is  very  convenient  for  use.  in 
connection  with  observations  of  the  sun,  but  owing  to 
the  irregularities  in  the  sun's  motion,  above  noted, 
apparent  solar  days,  hours,  etc.,  are  of  variable  length, 
a  day  in  December  being  nearly  a  minute  longer  than  one 
in  September.  When  time  is  to  be  kept  by  an  accurately 
constructed  clock  or  watch  such  irregularities  are  intoler- 
able, and  for  the  sake  of  clocks  and  watches  there  is 
employed  for  most  purposes  the  third  system,  viz., 

Mean  Solar  Time. — In  this  system  the  days  and  other 
units  are  of  uniform  length  and  equal,  respectively,  to  the 
mean  length  of  the  corresponding  units  of  apparent  solar 
time.  The  zero  point  of  the  system  is  an  imaginary- 
body,  called  the  mean  sun,  that  is  supposed  to  move 
uniformly  along  the  equator,  keeping  as  nearly  in  the 
same  right  ascension  with  the  true  sun  as  is  consistent 
with  perfect  uniformity  of  motion.  The  mean  solar 
time  at  any  moment  is  the  hour  angle  of  the  mean  sun 


40  FIELD  ASTRONOMY. 

and,  numerically,  it  differs  from  the  corresponding  true 
solar  time  by  the  difference  between  the  hour  angles, 
or  right  ascensions,  of  the  true  and  mean  suns.  This 
difference  is  called  the  equation  of  time  (the  ' '  sun  fast ' ' 
and  ' '  sun  slow  "  of  the  common  almanacs  and  calendars), 
and  its  value  for  each  day  of  the  year,  at  Greenwich  noon 
and  at  Washington  noon,  is  given  in  the  American 
Ephemeris  (see  §  21)  and  other  almanacs. 

To  change  local  solar  time  from  one  system  to  the  other 
we  have  therefore  to  interpolate  the  equation  of  time 
from  the  almanac,  with  the  argument  the  given  local 
time,  reduced  to  the  Greenwich  or  Washington  meridian 
by  addition  of  the  longitude,  and  apply  this  difference 
with  its  proper  sign  to  the  given  local  time.  For  exam- 
ple, let  it  be  required  to  find  for  the  meridian  of  Denver, 
and  for  the  date  May  10,  1905,  the  local  apparent  solar 
time  corresponding  to  the  mean  solar  time,  M,  given 
below.  The  course  of  the  computation  is  as  follows : 

A,  Denver  west  of  Greenwich 6h  59™  47*. 6           i 

M ,  Denver  Mean  Solar  Time 3      5     10.5          a 

Greenwich  Mean  Solar  Time 10      4     58  .1  J  +  2 

Equation  of  Time +3    44  . 2          3 

Denver  Apparent  Solar  Time 3      8     54.7  2  +  3 

19.  Relation  of  Sideral  to  Meaa  Solar  Time. — Once  in 
each  year  the  mean  sun  passes  through  the  vernal 
equinox  and  at  this  instant,  which  we  shall  represent  by 
V,  mean  solar  time  and  sidereal  time  agree.  At  any 
other  moment  during  the  year  sidereal  time  will  be 
greater  than  mean  solar  time  by  an  amount,  Q,  equal  to 
the  right  ascension  of  the  mean  sun  at  that  moment, 


TIME.  41 

and  the  conversion  of  the  one  kind  of  time  into  the 
other  is  therefore  solely  a  matter  of  finding  the  corre- 
sponding value  of  Q,  i.e., 

0  =  M+Q,  (18) 

where  0  and  M  stand  respectively  for  the  corresponding 
sidereal  and  mean  solar  time. 

Since  Q  increases  by  exactly  24h  in  one  mean  solar 
(tropical)  year  we  may  find  its  rate  of  increase,  5,  by 
dividing  24h  by  the  number  of  days  in  such  year 
(365.2422)  and  obtain  thus 

log  5  =  0.595781,  minutes  of  time  per  day, 
5  =  236.555,    seconds  of  time  per  day, 

and  since  the  motion  of  the  mean  sun  is  entirely  uni- 
form we  shall  have  at  any  time,  T,  the  value  of  Q  given 
by  the  relation 


The  quantity  V  varies  slightly  from  year  to  year  and 
its  value,  expressed  in  days  reckoned  from  the  first 
Greenwich  Mean  Noon  of  each  year  from  1905  to  1930,  is 
given  in  Table  2.  The  instant,  T,  must  be  similarly  ex- 
pressed for  use  in  Eq.  19.  Use  the  second  column  of 
Table  3  to  convert  ordinary  dates  into  days  from  the 
beginning  of  the  year. 

For  a  rough  computation  "in  the  head"  it  is  of  ten 
convenient  to  use 

5=4m(i-^r)         V=March  22.7  (Greenwich)      (20) 


42 


FIELD   ASTRONOMY. 


The  Nautical  Almanac  gives  for  each  day  of  the 
year  the  value  of  Q  at  Greenwich  Mean  Noon  (in  the 
last  column  of  page  II  ,of  the  monthly  calendar),  and 
also  gives  at  the  end  of  the  almanac,  Table  III,  a  table 
of  proportional  parts  by  which  to  facilitate  the  inter- 
polation of  Q  for  intermediate  times.  If  M  represents 
any  local  mean  solar  time  which  it  is  desired  to  convert 
into  sidereal  time  and  A  be  the  longitude  of  the  local 
meridian  west  of  Greenwich,  then  will  M  +  X  be  the 
corresponding  Greenwich  time  and  with  this  argument 
we  obtain  immediately  from  Table  III  the  correction 
that  added  to  the  tabular  value  of  Q  for  the  given  date 
furnishes  its  value  at  the  given  moment. 

For  illustration  of  each  process  above  given  for 
finding  Q  let  it  be  required  to  find  the  sidereal  time 
corresponding  to  Boston  mean  solar  time  ph  i9m  2 6s. 6 7 
on  August  4,  1905,  assuming  as  the  longitude  of  Boston 
4h  44m  i5s  west  of  Greenwich. 


First  Method. 

Second  Method. 

h.     m.         s. 

h.  m.    s. 

M 

9     19     26.67 

M 

9   19   26.67 

A 

4     44     15 

A 

4  44  15 

M  +  A 

14       3     42 

M  +  A 

14     3  42 

T 

217.586 

Greenwich,  Q 

8  49  31.43 

V 

82.691 

Table  III 

2  18.60 

log(r-y) 

2.129996 

Local,  Q 

8  51  50.03 

log  S 

0.595781 

6 

18  ii  16.70 

S(T-V) 

531.835  minutes 

Q 

8     51     50.10 

e 

18     ii     16.77 

The  difference  in  the  results  here  found  for  M  is  due 
to  neglected  figures  in  the  fourth  decimal  place  of  V  and 
T  and  illustrates  the  superior  precision  as  well  as  con- 
venience of  the  second  method.  The  latter  should 


TIME.  43 

always  be  employed  when  an  almanac  for  the  current 
year  is  available,  but  the  first  method  may  render  good 
service  in  the  absence  of  such  an  almanac. 

The  converse  problem  of  transforming  sidereal  time 
into  mean  solar  time  is  best  treated  as  follows  :  Let  M\ 
denote  an  assumed  mean  solar  time  not  very  different 
from  the  true  value  of  M  corresponding  to  the  given  0 
and  compute,  as  above,  the  corresponding  values  of  Q 
and  0  which  we  represent  by  Q\  and  Q\.  If  Q\  proves  to 
be  different  from  the  given  0,  Qi  will  differ  from  the 
true  Q  by  the  amount  that  the  mean  sun  changes  its 
right  ascension  in  the  interval  0—0l  ,  i.e.,  the  true  value 
of  Q  will  be 

0i),  (21) 


when  6—  6  1  is  expressed  in  sidereal  minutes.  When 
values  of  Q  are  taken  from  the  Almanac  it  is  customary 
to  adopt  the  local  mean  noon  as  the  value  of  M\  since 
in  this  case  the  value  of  M  becomes 

M=(0-Qi)-oM64(tf-Qi).  (22) 

\ 

The  Qi  thus  defined  is  obtained  by  adding  to  the 
Almanac  value  of  Q  a  correction  taken  from  Table  III 
with  the  observer's  longitude  as  argument,  and  the  last 
term  of  the  expression  is  taken  from  a  table  of  multiples 
of  os.  1  64  given  at  the  end  of  the  Almanac  as  Table  II. 
In  the  absence  of  the  Almanac  it  is  better  to  take  the 
assumed  Mi  as  near  as  may  be  to  the  true  value  of  M. 

Using  the  Almanac  we  reconvert  the  0  found   above 
into  the  corresponding  mean  solar  time  as  follows  : 


44  FIELD    ASTRONOMY. 


Greenwich,  Q 
Table  III,  X 

8  49  3i-43 
46.70 

Local  Noon,  Ql 

8  50  18.13 

6 

18  ii   16.70 

0-Qi 

9  20  58.57 

Table  II,  0-^ 

i  31.90 

M 

9  19  26.67 

Without  the  aid  of  an  Almanac,  we  may  assume  Mi  =  9h 
and  by  a  process  entirely  similar  to  the  First  Method 
employed  above  find  the  Qi  shown  in  the  following 
schedule  and  determine  from  it  the  required  M  as 
follows ; 


Ml 

°d 
Q 

M 


900 
8  51  46.80 

17  51  46.80 

18  ii   16.70 

8  51  49-99 

9  19  26.71 


20.  Chronometer  Corrections.  —  As  already  indicated, 
in  actual  practice  the  measurement  of  time  is  made  by 
clocks  or  chronometers. 

A  chronometer  does  not  differ  essentially  from  an 
ordinary  watch,  and  like  the  latter  is  designed  to  show 
upon  its  face,  at  each  moment,  the  mean  solar  time  (or 
sidereal  time)  of  some  definite  meridian,  e.g.,  the  merid- 
ian 90°  west  of  Greenwich.  Since  the  time  indicated 
by  such  an  instrument  is  seldom  correct,  the  error  of  the 
timepiece  must  usually  be  taken  into  account,  and  in 
astronomical  practice  this  is  done  through  the  equation, 


T,  or    M  =  T  +  JT,  (24) 

where   T  is  the  time   shown  by  the  chronometer   (or 
watch)  and   AT  is  the    correction  of  the  chronometer, 


TIME.  45 

i.e.,  the  quantity  which  must  be  added,  algebraically, 
to  the  watch  time  in  order  to  obtain  true  time  of  the 
given  meridian.  When  the  chronometer  is  too  slow  T 
is  less  than  the  true  time  at  any  moment,  and  AT  is  there- 
fore positive  in  this  case  and  negative  when  the  chro- 
nometer is  too  fast.  While  the  symbol  AT  always  repre- 
sents a  chronometer  correction  its  numerical  value  in  a 
given  case  depends  upon  the  particular  use  required, 
i.e.,  whether  the  chronometer  time  is  to  be  reduced  to 
sidereal,  or  solar,  local,  or  standard  time.  In  the  two 
Equations  24,  therefore,  AT  represents  quite  different 
quantities,  since  6  and  M  are  usually  different  one  from 
the  other,  and  in  every  case  a  special  memorandum 
must  be  made  showing  whether  the  given  AT  relates  to 
sidereal,  mean  solar,  or  apparent  solar  time. 

If  the  chronometer  gains  or  loses,  it  is  said  to  have  a 
rate  and  AT  will  then  change  from  day  to  day.  If  we 
assume  a  uniform  rate,  the  relation  between  T  and  0 
becomes 

0  =  T  +  AT9  +  P(T-TJ,  (25) 

where  the  subscript  0  denotes  the  particular  value  of  AT 
belonging  to  the  chronometer  time  T0,  and  p  is  the  rate 
of  the  chronometer  per  day  or  per  hour,  positive  when 
the  chronometer  is  losing  time.  The  interval  T—T0 
must  be  expressed  in  the  same  unit  as  that  for  which  p 
is  given,  hours  for  an  hourly  rate,  etc.  A  similar  equa- 
tion represents  the  relation  between  T  and  M,  but  p  and 
AT  will  be  numerically  different  from  the  values  required 
in  Equation  25. 


46  FIELD    ASTRONOMY. 

A  sidereal  chronometer,  i.e.,  one  intended  to  keep 
sidereal  time,  differs  from  a  mean  solar  chronometer 
only  in  the  more  rapid  motion  of  its  mechanism,  and  is 
in  fact  an  ordinary  timepiece  for  which  p=—  3m  56S«5 
per  day.  Similarly  a  watch  may  be  regarded  as  a 
sidereal  timepiece  for  which  /?=  +  ios  per  hour.  A 
sidereal  chronometer  is  most  convenient  for  use  in  obser- 
vations of  stars,  since  their  diurnal  motion  in  hour  angle 
is  proportional  to  the  lapse  of  sidereal  time,  but  these 
observations  may  perfectly  well  be  made  with  a  watch 
or  other  mean  solar  timepiece,  provided  this  is  treated 
as  a  sidereal  chronometer  with  a  large  rate;  e.g.,  if  p 
denote  the  hourly  rate  of  the  watch  relative  to  mean  solar 
time,  its  hourly  rate  upon  sidereal  time  will  be  p'  =  p-\-  ios. 
Use  Equation  25  in  connection  with  this  value  of  pf 
to  determine  from  minute  to  minute  the  varying  value 
of  JT. 

21.  The  Almanac.  —  The  American  Ephemeris  and 
Nautical  Almanac  is  an  annual  volume  issued  by  the 
U.  S.  Navy  Department  for  the  use  of  navigators,  astron- 
omers, and  others  concerned  with  astronomical  data. 
These  data  are  for  the  most  part  quantities  that  vary 
from  day  to  day  and  whose  numerical  values  are  given 
at  convenient  intervals  of  Greenwich  or  Washington 
boicti  time,  e.g.,  the  E  and  Q  of  the  preceding  sections, 
and  the  right  ascensions  and  declinations  of  the  sun, 
moon,  planets,  and  principal  fixed  stars.  The  varia- 
tions of  these  quantities  are  due  to  many  causes,  orbital 
motion,  precession,  nutation,  aberration,  etc.,  that,  in 
general,  lie  beyond  the  scope  of  the  present  work,  but 


TIME.  47 

we  shall  have  frequent  occasion,  as  in  §§  18  and  19,  to 
take  from  the  almanac  numerical  values  of  the  quanti- 
ties above  indicated,  and  these  values  are  to  be  inter- 
polated for  some  particular  instant  of  time,  usually  that 
of  an  observation  in  connection  with  which  they  are 
required,  as  logarithms  are  interpolated  to  correspond 
to  some  particular  value  of  the  argument  of  the  table. 
Since  quantities  are  tabulated  in  the  almanac  for  selected 
instants  of  Greenwich  or  Washington  time,  the  time  used 
as  the  argument  for  their  interpolation  must  be  referred 
to  one  of  these  meridians  (see  §  17). 

For  a  detailed  account  of  the  way  in  which  the  almanac 
is  to  be  used,  consult  the  explanations  given  at  the  end  of 
each  volume,  under  the  title,  Use  of  the  Tables.  In 
addition  to  those  explanations  it  should  be  noted  that 
under  the  heading  Fixed  Stars,  pages  304-399,  there 
are  given  three  separate  tables,  from  the  last  of  which, 
bearing  the  subtitle  Apparent  Places  for  the  Upper 
Transit  at  Washington,  accurate  coordinates  of  most  of 
the  stars  may  be  obtained  for  use  in  the  reduction  of 
observations.  For  the  remaining  stars,  five  in  number 
and  all  very  near  the  celestial  pole,  special  provision  of 
this  kind  is  made  in  the  second  table,  which  bears  the 
subtitle  Circumpolar  Stars.  Look  here  for  the  coor- 
dinates of  Polaris.  The  first  table,  under  the  subtitle 
Mean  Places,  etc.,  gives  in  very  compact  form,  for  all 
stars  contained  in  the  other  two  tables,  right  ascensions 
and  declinations,  together  with  their  Annual  Variations, 
that  may  be  consulted  with  advantage  when  only  approx- 


48  FIELD    ASTRONOMY. 

imate  values  of  these  quantities  are  required,  e.g.,  in  the 
preliminary  selection  of  stars  suitable  to  be  observed. 

In  this  connection  the  second  column  of  the  table 
of  Mean  Places,  entitled  Magnitude,  deserves  especial 
notice,  since  it  furnishes  an  index  to  the  brightness  of 
the  stars,  which  is  an  important  element  in  deciding 
upon  their  availability  for  a  given  instrument.  The 
brightness  of  each  star  is  represented  by  a  number 
adapted,  upon  an  arbitrary  scale,  to  that  brightness, 
so  that  a  very  bright  star  is  represented  by  the  number 
o,  one  at  the  limit  of  naked  eye  visibility  by  6, 
and  intermediate  degrees  of  brightness  are  represented 
by  the  intermediate  numbers,  carried  to  tenths  of  a  mag- 
nitude. Polaris  is  of  the  magnitude  2.2,  and  is  a  con- 
spicuous object  in  even  a  very  small  telescope,  provided 
the  telescope  is  properly  focussed.  In  the  telescope 
of  an  engineer's  transit,  stars  of  the  magnitude  4.0  or 
even  5.0  may  be  readily  observed,  while  with  a  sextant, 
under  ordinary  conditions,  the  third  magnitude  may 
be  taken  as  the  limit  of  availability. 


CHAPTER  IV. 
CORRECTIONS  TO   OBSERVED   COORDINATES. 

IT  has  already  been  pointed  out  that  the  problems 
of  spherical  astronomy  are  in  great  part  cases  of  the 
transformation  of  coordinates  between  systems  having 
a  common  origin  but  different  axes,  and  it  should  be 
noted  that  the  observed  data  for  these  transformations 
frequently  require  some  correction  before  they  can  be 
introduced  into  the  equations  furnished  by  the  astro- 
nomical triangle.  Aside  from  errors  arising  from  de- 
fective adjustment  or  other  purely  instrumental  causes, 
the  observed  coordinates  of  a  celestial  body  may  require 
any  or  all  of  the  following  corrections. 

22.  Dip  of  the  Horizon.  —  This  correction  is  required 
when  an  altitude  is  to  be  derived  from  a  measurement 
of  the  angle  of  elevation  of  a  body  above  the  sea  horizon. 
Owing  to  the  spherical  shape  of  the  earth  the  visible 
sea  horizon  always  lies  below  the  plane  of  the  observer's 
true  horizon,  and  the  amount  of  this  depression  might 
easily  be  determined  from  the  geometrical  conditions 
involved,  were  it  not  that  the  rays  of  light  coming  to 
the  observer  from  near  the  horizon  are  bent  by  the  at- 
mosphere (refraction),  in  a  manner  that  does  not  admit 

49 


50  FIELD  ASTRONOMY. 

of  accurate  estimation  in  any  given  case,  although  its 
average  amount  is  fairly  well  known.  We  therefore 
abstain  from  any  formal  investigation  of  this  correc- 
tion, and  expressing  by  e,  in  feet,  the  observer's  elevation 
above  the  water,  we  adopt  as  a  sufficient  approximation 
to  the  observed  amount  of  the  depression,  either  of  the 
following  formulae, 

^-"sA—rW',       £>"  =  [i.7738]V*.  (26) 

The  values  of  D  given  by  these  equations  are  expressed 
in  minutes  and  seconds,  respectively,  but  owing  to  varia- 
tions in  the  amount  of  the  refraction  the  numerical 
values  furnished  in  a  given  case  may  be  in  error  by 
several  per  cent.  As  a  correction  D  must  always  be 
so  applied  as  to  diminish  the  observed  elevation  above 
the  horizon. 

Note  that  if  the  depression  of  the  visible  horizon 
be  measured  with  a  theodolite  or  other  suitable  instru- 
ment, Equation  26  will  furnish  an  approximate  value 
of  the  elevation  of  the  instrument  above  the  water. 

23.  Refraction.  —  In  general  the  apparent  direction 
of  a  star  is  not  its  true  direction  from  the  observer,  since 
the  light  by  which  he  sees  it  has  been  bent  from  its 
original  course  in  passing  through  the  earth's  atmos- 
phere. The  resulting  displacement  of  the  star  from  its 
true  position  is  called  refraction,  and,  like  the  similar 
effect  noted  in  the  previous  section,  its  analytical  treat- 
ment presents  mathematical  and  physical  problems 
whose  solution  must  be  sought  in  more  advanced  works 
than  the  present.  Some  of  the  results  of  that  solution 


CORRECTIONS  TO  OBSERVED  COORDINATES.          51 

which  we  shall  have  occasion  to  use  hereafter  are  as 
follows:  Save  at  very  low  altitudes,  less  than  10°,  the 
refraction  does  not  sensibly  change  the  azimuth  of  a 
star,  but  its  whole  effect  is  to  increase  the  altitude,  so 
that  every  star  appears  nearer  to  the  zenith  than  it  would 
appear  if  there  were  no  refraction.  The  amount  of  this 
displacement  depends  chiefly  upon  the  star's  distance 
from  the  zenith,  but  is  also  dependent  in  some  measure 
upon  the  temperature  of  the  air  and  its  barometric 
pressure. 

If  we  represent  by  h'  the  star's  altitude  as  measured, 
by  h  the  corresponding  true  altitude,  by  t  the  tempera- 
ture, in  degrees  Fahrenheit,  of  the  air  surrounding  the 
observer,  and  by  B  the  barometric  pressure  in  inches, 
the  amount  of  the  refraction  in  seconds  of  arc  may  be 
obtained  from  the  following  formula  within  half  a 
second  for  all  altitudes  greater  than  20°, 

(27) 


For  altitudes  less  than  20°  this  formula  gives  results 
that  are  too  great  and  as  the  star  approaches  the  hori- 
zon the  error  increases  rapidly,  as  may  be  seen  from 
the  following  brief  table  of  corrections  required  to 
reduce  the  numbers  given  by  Eq.  2  7  to  the  true  values 
of  the  refraction. 


h 

Corr. 

5° 

10 

15 

20 

25 

-68" 

-  9 

—     2 
—    0 
+    0 

The  readings   of  a  mercurial  barometer,   B',  do  not 


52  FIELD  ASTRONOMY. 

furnish  immediately  the  barometric  pressure,  B,  but 
require  a  "reduction  to  the  freezing-point,"  i.e.,  a  cor- 
rection to  reduce  the  reading  to  what  it  would  be  if  the 
mercury  were  at  the  normal  temperature  assumed  in  the 
theory  of  the  barometer.  This  reduction  may  be  ob- 
tained with  sufficient  accuracy  from  the  equation 


(28) 


10  000 


where  T  is  the  temperature  of  the  mercury,  in  degrees 
Fahr.,  and  the  barometer  reading  and  its  resulting  cor- 
rection are  expressed  in  inches. 

24.  Semi-  diameter.  —  Observations  of  the  sun  or  other 
body  presenting  a  sensible  disk  are  usually  made  by 
pointing  the  instrument  at  the  edgo  of  the  body,  techni- 
cally called  the  limb,  and  the  resulting  altitude  or  azimuth 
is  that  of  the  limb  observed,  while  the  data  furnished 
by  the  almanac  relate  to  the  centre  of  the  body.  The 
semi-diameters  of  the  sun,  moon,  and  planets,  i.e.,  the 
angles  subtended  at  the  earth  by  their  respective  radii, 
are  given  in  the  almanac  at  convenient  intervals  of  time, 
and  the  interpolated  values  of  these  quantities  may  be 
used  to  pass  from  the  observed  coordinates  of  the  limb 
to  those  of  the  centre  of  the  body,  e.g.,  the  sun.  In 
the  case  of  the  altitude  or  zenith  distance  we  have  the 
very  simple  relation 

h'=h"±S,  (29) 

where  5  denotes  the  semi-diameter  and  h"  and  hf  are,  re- 
spectively, the  observed  and  the  corrected  altitude.  The 


CORRECTIONS  TO  OBSERVED  COORDINATES. 


53 


sign  of  5  depends  upon  whether  the  lower  or  the  upper 
limb  was  observed. 

In  the  case  of  an  azimuth  the  relation  is  more 
complicated.  From  the  right-angled  spherical  triangle 
formed  by  the  zenith,  the  sun's  centre  and  that  point  of 
the  limb  at  which  the  latter  is  tangent  to  a  vertical  circle 
(see  Fig.  4)  we  obtain, 


sin  z  sin  (Ar  -  A")  =  sin  5, 


(30) 


which  determines  the  correction,  Af  —  A" ,  for  difference 
of  azimuth  between  centre  and  limb.     Since  5  does  not 
much  exceed    15',  we  may  in  most 
cases  assume  the  arcs  to  be  propor- 
tional to  their  sines  and  simplify  this 
rigorous  equation  to  the  form, 

A'=A"±S  seek,  (31) 

in  which  the  positive  sign  is  to  be 
used  for  the  following  and  the  nega- 
tive for  the  preceding  limb. 

25.  Parallax.  —  In  the  reduction 
of  astronomical  observations  it  is 
usually  necessary  to  combine  the 


cd  coordinates,    azimuth,  alti- ., 

FIG.  4. — Semi-diameter. 

tude,  etc.,  with  data   obtained  from 
the    almanac,   e.g.,   the    right    ascension    and    declina- 
tion of  the  body  observed.     But  the  origin  to  which 
these  latter  coordinates  are  referred  is  the  centre  of  the 
earth,  while  the  origin  for  the  observed  coordinates  is 


5±  FIELD  ASTRONOMY. 

at  the  eye  of  the  observer,  and  before  combining  these 
heterogeneous  data  we  must  reduce  them  to  a  common 
origin,  for  which  we  select  that  used  in  the  almanac. 

In  Fig.  5  let  C  represent  the  centre  of  the  earth,  0 
the  observer's  position,  and  P  the  observed  body,  at  the 
respective  distances  p  and  r  from  C.  Neglecting  the 


FIG.  5. — Parallax. 

earth's  compression,  i.e.,  its  slight  deviation  from  a  truly 
spherical  form,  the  line  OC  is  the  observer's  vertical 
and,  therefore,  OPC  is  a  vertical  plane  and  marks  out 
upon  the  celestial  sphere  a  vertical  circle,  against  which 
the  body  P  will  appear  projected  whether  seen  from  0 
or  C.  Its  azimuth  will,  therefore,  be  the  same  for  the 
two  origins  and  requires  no  reduction  to  the  centre  of  the 
earth. 

The  altitude,  however,  does  require  such  a  reduction, 
and  to  determine  its  amount  we  let  OH  in  Fig.  5  repre- 
sent the  plane  of  the  observer's  horizon  and  obtain  as 


CORRECTIONS   TO  OBSERVED  COORDINATES.          55 

the  observed  altitude  of  P  the  angle  there  marked  hf. 
As  seen  from  the  centre  of  the  earth  the  altitude  of  P 
will  be  measured  by  the  angle  PIH,  marked  h  in  the 
figure,  and  from  principles  of  elementary  geometry  we 
have 


This  last  angle  is  called  the  parallax  in  altitude,  and  rep- 
resenting it  by  P  we  find  from  the  triangle  OPC 

p  sin  (90°  +  h')=r  sin  P. 

Since  r  is  always  much  greater  than  p,  P  must  be  a  small 
angle  and,  applying  the  principles  of  §  4,  we  may  write, 
in  place  of  the  preceding  equation, 

P=h-h'  =  206265  —  cos  h',  (32) 

which  is  the  required  correction  to  reduce  an  observed 
altitude  to  the  corresponding  coordinate  referred  to  an 
origin  at  the  centre  of  the  earth. 

For  the  fixed  stars  this  correction  is  absolutely  insen- 
sible, less  than  o".oooi,  on  account  of  their  great  distance 
from  the  earth.  For  the  sun  and  planets  it  amounts  to  a 
few  seconds  of  arc,  and  in  its  computation  the  value  of  the 

coefficient,  206265  —  ,  should  be  taken  from  the  almanac, 

where  it  is  given  for  each  of  these  bodies  and  is  called 
their  horizontal  parallax,  since  it  is  the  amount  of  the 
parallax  in  altitude  when  the  body  is  in  the  horizon, 
hr  =  o°.  For  the  sun  it  is  usually  sufficient  to  assume 


56 


FIELD  ASTRONOMY. 


8".  8  as  a  constant  value  for  its  horizontal  parallax. 
The  moon's  parallax  is  much  greater,  about  i°,  and  the 
simple  analysis  given  above  neglects  some  factors  that 
are  of  sensible  magnitude  in  this  case,  although  for  ordi- 
nary purposes  they  may  be  ignored  in  connection  with 
every  other  celestial  body. 

Since  the  effect  of  parallax  is  to  make  the  body  appear 
farther  from  the  zenith  than  it  really  is,  the  corrections 
for  parallax  and  refraction  will  always  have  opposite 
signs. 

26.  As  an  example  of  such  corrections  and  their 
proper  sequence  we  give  the  following  reduction  of  an 
angle  observed  between  the  sun's  upper  limb  and  the 
sea  horizon,  as  seen  by  an  observer  at  an  elevation  of 
63  feet  above  the  water. 


Date 

Oct.  6,  1907 

Refraction  Const. 

2.989 

Temperature 

44°  Fahr. 

B 

1.466 

Barometer 

29.26  in. 

colog  (456  +  0 

7.301 

B'-B 

.04 

cot  h' 

0.282 

B 

29.22 

logR 

2.038 

Observed  Angle 

27°  43'  20" 

R 

109" 

h" 

27    33    39 

Dip  of  Horizon 

Semi  -Diameter 

16      9 

e 

63 

V" 

27     17    37 

Constant 

1-774 

Parallax  Const. 

0.944 

log  \/e 

0.900 

cos  h'" 

9-949 

Dip 

472"  = 

Parallax 

8" 

Dip 

7'  52" 

h 

27     17     45 

V 

27    35    28 

. 

All  computations  such  as  the  above  may  conveni- 
ently be  made  with  a  slide  rule. 

In  accordance  with  general  custom  the  symbol  log 
is  printed  in  the  above  schedule  only  when  necessary  to 


CORRECTIONS    TO  OBSERVED  COORDINATES.          57 

avoid  misunderstanding,  as  at  the  bottom  of  the  first 
column.  Usually  the  figures  themselves  indicate  whether 
they  are  logarithms  or  natural  numbers;  e.g.,  the  several 
numbers  marked  Const,  are  clearly  the  logarithms  of 
constant  coefficients.  For  similar  reasons  of  convenience 
the  —  10  that  strictly  should  be  placed  after  a  logarithm 
whose  characteristic  has  been  increased  by  10  is  usually 
left  to  the  imagination. 

27.  Diurnal  Aberration.  —  There  is  a  very  small  cor- 
rection to  observed  data,  arising  from  the  fact  that  the 
observer  himself  is  not  at  rest  relative  to  the  stars,  but 
is  always  in  rapid  motion  toward  the  east  point  of  his 
horizon,  carried  along  by  the  earth  in  its  diurnal  rotation. 
This  correction  is  so  small  that  it  may  usually  be  omitted 
and  we  therefore  abstain  from  an  analytical  investigation 
of  its  effect,  such  as  may  be  found  in  the  larger  treatises 
upon  spherical  astronomy,  and  note  as  a  result  of  that 
investigation  that  all  stars  when  near  the  meridian  are 
displaced  toward  the  east  point  of  the  horizon  through 
an  angular  distance  equal  to  o".32  cos  0,  where  0  de- 
notes the  observer's  latitude.  As  a  result  of  this  dis- 
placement each  star  comes  a  little  later  to  the  meridian 
than  it  otherwise  would  come  and  since  the  rate  of  motion 
of  a  star  when  measured  in  arc  of  a  great  circle  is  propor- 
tional to  the  cosine  of  its  declination,  the  amount  of  this 
retardation,  expressed  in  time,  is  os.o2i  cos  0  sec  d. 
See  the  theory  of  the  transit  instrument  for  an  example 
of  the  application  of  this  correction,  and  see  also  the  de- 
termination of  precise  azimuths  for  another  case  in  which 
diurnal  aberration  is  to  be  taken  into  account. 


CHAPTER  V. 

ROUGH     DETERMINATIONS    OF  TIME,  LATITUDE,  AND 

AZIMUTH. 

28.  General  Considerations.  —  For  the  purposes  of 
field  astronomy,  which  are  the  only  ones  contemplated 
in  the  present  work,  the  most  important  astronomical 
problems  relate  to  the  determination  of  time,  latitude, 
and  azimuth. 

A  time  determination  implies  the  making  and  reduc- 
ing of  astronomical  observations  which  suffice  to  furnish 
the  correction,  AT,  of  a  chronometer  or  other  timepiece, 
and  for  this  purpose  we  obtain  from  §§  15  and  20  the 

relations 

a  +  t  =  6  =  T  +  JT,  (33) 

where  a  and  /  represent  the  right  ascension  and  hour  angle 
of  any  star  at  the  chronometer  time  T.  The  student 
should  particularly  note  that  the  chronometer  is  not 
supposed  to  be  correctly  set  ;  T  is  the  time  shown  by  the 
chronometer  regardless  of  whether  that  time  be  right  or 
wrong,  since  the  AT  fully  compensates  for  any  error  of 
this  kind.  In  the  case  of  the  sun  we  have,  from  the  rela- 
tion between  mean  and  apparent  solar  time, 

(34) 


where  E  denotes  the  equation  of  time  at  the  instant  T. 

58 


ROUGH    DETERMINATIONS   OF    TIME,  ETC.  59 

Since  a  and  E  may  be  obtained  from  the  almanac,  any 
observation  which  determines  the  hour  angle  of  a  celestial 
body  at  the  observed  time  T  will  suffice  to  determine 
4T,  and  such  an  observation  when  properly  reduced 
constitutes  a  time  determination. 

An  azimuth  determination  may  be  required  either 
for  fixing  the  true  azimuth  of  the  line  joining  two  terres- 
trial points,  or  for  determining  the  relation  of  a  particular 
instrument  to  the  meridian;  e.g.,  to  determine  the  read- 
ing, K,  to  which  the  azimuth  circle  of  a  theodolite  must 
be  set,  in  order  that  the  line  of  sight  shall  point  due  south. 
A  theodolite  is  said  to  be  oriented  when  its  verniers  have 
been  set  to  read  the  true  azimuth  of  the  object  toward 
which  the  line  of  sight  is  directed,  i.e.,  when  K  =  o. 

By  a  latitude  determination  we  mean  any  set  of  ob- 
servations from  which  a  knowledge  of  the  observer's 
latitude  may  be  obtained. 

For  each  of  these  determinations,  time,  azimuth, 
latitude,  many  methods  have  been  devised  and  these 
differ  greatly  among  themselves  with  respect  to  the 
instrumental  equipment  and  expenditure  of  time  and 
labor  which  they  require,  and  with  respect  to  the  corre- 
sponding degree  of  accuracy  furnished  in  their  results. 
In  any  given  case  a  choice  must  be  made  among  these 
methods  with  reference  to  the  required  precision  of  the 
results  and  also  with  reference  to  convenience  and 
economy  in  obtaining  it.  To  facilitate  this  choice  the 
methods  to  be  presented  in  the  following  pages  are 
classified  as: 

(A)  Rough  Determinations;  in  which  there  may  be 


60  FIELD   ASTRONOMY. 

permitted  in  the  final  result  an  error  amounting  to  two 
minutes  of  arc  or  one  tenth  of  a  minute  of  time. 

(B)  Approximate  Determinations;  in  which  the  final 
errors  ought  not  to  exceed  15"  and  is  respectively. 

(C)  Accurate  Determinations;  in  which  the  required 
precision  is  limited  only  by  the  capacity  of  the  instru- 
ment and  of  the  observer.    In  the  case  of  a  sextant  this 
limit  may  be  placed  at  2"  or  3",  and  for  a  good  engineer's 
transit  at  i".     We  proceed  first  to  consider  that  class 
of    observations  whose    advantage  consists  in  economy 
of  time  and  labor,  viz.,  rough  determinations. 

29.  Latitude.  —  A  determination  of  any  one  of  the 
quantities  time,  latitude,  or  azimuth  is  greatly  facilitated 
by  a  knowledge  of  one  or  both  of  the  others,  and  if  all 
three  are  unknown,  the  simplest  mode  of  procedure  is 
to  observe  the  Pole  Star  as  set  forth  in  §32.  But  this 
commonly  requires  observations  by  night,  which  may 
be  inconvenient,  and  by  day  the  sun  is  the  object  most 
readily  available. 

From  the  astronomical  triangle,  or  from  Equations  14, 
it  is  apparent  that  when  the  sun  is  on  the  meridian, 
i.e.,  when  t  =  o,  its  altitude  is  a  maximum,  and  if  this 
maximum  altitude  be  measured  with  a  sextant  or  theod- 
olite it  will  furnish  a  latitude  determination  through  the 
equation,  true  for  noon  only, 

0  =  8  +  z  =  9o°  +  d-h,  (35) 

which  may  be  obtained  by  inspection  from  Fig.  2,  or 
analytically  from  the  last  of  Equations  14.  With  the 
instrument  employed,  follow  the  sun's  motion  in  altitude 


ROUGH   DETERMINATIONS   OF    TIME,   ETC.  61 

until  it  begins  to  dimmish,  and  take  the  greatest  reading 
obtained  as  corresponding  to  the  maximum  altitude. 

This  reading,  or  the  altitude,  h',  derived  from  it,  will 
require  correction  for  instrumental  errors,  semi-diameter, 
etc.,  as  shown  in  Chapter  IV,  but  the  application  of  these 
corrections  may  be  abbreviated  by  interpolating,  in 
minutes  of  arc,  the  combined  correction  for  refraction  and 
parallax  from  Table  I,  at  the  end  of  the  book. 

This  gives,  with  the  observed  altitude  as  argument 
under  the  heading  R,  the  amount  of  the  refraction  cor- 
responding to  an  average  condition  of  the  atmosphere, 
viz.,  Bar.  29.0  in.,  Temp.  50°  Fahr.,  while  under  the 
heading  R'  is  given  the  combined  effect  of  refraction 
and  parallax  for  the  sun.  Use  these  tables  for  the 
reduction  of  any  altitude  of  sun  or  star  when  the  re- 
quired precision  is  not  greater  than  10".  This  table 
may  be  adapted  even  to  extreme  atmospheric  condi- 
tions by  increasing  R  or  R'  by  one  per  cent  for  each 
0.3  inch  that  the  barometric  pressure  exceeds  29  inches. 
Similarly  diminish  R  by  one  per  cent  for  each  5°  that 
the  temperature  exceeds  50°  Fahr. 

The  following  latitude  determination  was  made  by 
measuring  with  a  sextant  and  artificial  horizon  (§59) 
the  maximum  double  altitude  of  the  sun's  lower  edge 
(limb)  upon  a  date,  Dec.  19,  1898,  at  which  the  sun's 
declination,  as  furnished  by  the  almanac,  was,  d  = 
—  23°  26'.o.  The  reduction  of  the  observation  is  a? 
follows : 


62  FIELD  ASTRONOMY. 

Sextant  Reading  57°  44'  30" 
Instrumental  Corr.        —  i    50 

Corr'd  Sextant  57    42    40 

.h'  28    51.3 

Ref.-Par.,  R'        -    1.6 

Semi-diameter  +16.3 

h  296.0 

90°  +  8  66   34.0 

Latitude,  (f>  37   28.0 

Make  a  determination  of  your  own  latitude  by  a  simi- 
lar method. 

30.  Time  and  Azimuth  from  an  Observed  Altitude. — 
If  the  latitude  be  thus  observed  at  noon,  time  may  be 
determined  with  a  sextant,  and  both  time  and  azimuth 
may  be  determined  with  a  theodolite  by  measuring  an 
altitude  of  the  sun  when  at  a  considerable  distance  from 
the  meridian.  Any  error  in  the  assurned  latitude  or  in 
the  measured  altitude  will  affect  both  the  azimuth  and 
time  resulting  from  the  observation,  but  the  influence 
of  these  errors  may  be  greatly  reduced  by  observing 
the  sun  when  near  the  prime  vertical  and;  for  azimuth, 
when  at  a  low  altitude,  e.g.,  10°  or  15°. 

When  a  theodolite  is  used,  there  should  be  at  least 
two  observations  of  altitude  made,  one  Circle  R.  and 
the  other  Circle  L.,  in  order  to  eliminate  instrumental 
errors  (see  §  50).  Observe  the  edges  of  the  sun,  not 
its  center,  and  correct  the  results  for  semi-diameter 
'(§  24);  but  if  the  instrument  has  stadia  threads  less 
than  30'  apart,  avoid  this  correction  as  follows:  Point 
the  telescope  at  the  sun  so  that  the  two  horizontal 
threads  cut  off  equal  segments  from  the  upper  and  lower 


PLATE  I. 


\Tofacep.  62.] 


An  American  Engineer's  Transit.     Diameter  of  Horizontal  Circle  7  inches. 
Approximate  Cost  $350. 

~ 

i£ 


ROUGH    DETERMINATIONS    OF    TIME,    ETC.  63 

edges  of  the  sun,  and  by  turning  the  slow-motion  screw 
in  altitude,  keep  these  segments  of  equal  area  as  the  sun 
drifts  across  the  field  of  view,  until  it  reaches  a  position 
in  which  the  vertical  thread  bisects  each  segment. 
Record  this  time  to  the  nearest  second,  and  also  record 
the  readings  of  the  four  verniers  of  the  instrument. 
Before  reversing  the  instrument  to  obtain  the  second 
observation,  read  and  record  both  its  levels  (azimuth 
and  altitude  levels,  §§48  and  50),  and  after  reversing 
bring  the  bubbles  back,  by  means  of  the  levelling-screws, 
to  the  position  thus  recorded.  This  process  eliminates 
errors  of  level. 

The  better  class  of  engineer's  transits  are  usually 
provided  with  shade-glasses  to  moderate  the  intensity 
of  the  sun's  light  and  permit  it  to  be  viewed  through 
the  telescope.  But  these  glasses  are  by  no  means  neces- 
sary, since  an  image  of  the  sun  and  the  threads  of  the 
instrument  may  be  projected  upon  a  piece  of  cardboard 
and  be  there  seen  and  observed  quite  as  accurately  as 
in  the  telescope.  Pull  the  eyepiece  out,  away  from 
the  threads,  until  the  latter  can  no  longer  be  seen  dis- 
tinctly with  the  eye ;  then  allow  the  sun  to  shine  through 
the  telescope  upon  the  cardboard  held  behind  the  eye- 
piece, and  shift  the  cardboard  toward  and  from  the 
instrument  until  a  position  is  found  in  which  the  projected 
images  of  the  threads  appear  sharp  and  distinct.  Then 
turn  the  focussing-screw  until  the  edge  of  the  sun's 
image  also  appears  well  defined  and  the  projected  images 
will  be  ready  for  observation.  To  diminish  as  far  as 
possible  the  effect  of  errors  in  the  measured  altitude 


64  FIELD   ASTRONOMY. 

and  the  latitude  assumed  in  the  reduction,  observe  the 
sun  when  it  is  as  near  as  may  be  due  east  or  west,  but 
its  altitude  should  not  be  less  than  10°. 

Reduction  of  the  Observations.  —  Observations  con- 
ducted as  in  the  following  record  will  furnish  as  data 
the  true  altitude  of  the  sun's  centre  and  the  horizontal 
angle,  H,  between  the  sun's  centre  and  some  fixed  object 
whose  azimuth  is  to  be  determined,  both  quantities  cor- 
responding to  one  and  the  same  time,  T,  taken  from 
some  watch  or  other  time  piece.  These  data  must  be 
supplemented  by  the  latitude,  <£,  supposed  to  be  inde- 
pendently known,  and  by  the  sun's  declination,  d,  and 
equation  of  time,  E,  to  be  obtained  from  the  almanac. 
Reference  to  Fig.  3  will  show  that  d,  h  and  (j>  are  respec- 
tively the  complements  of  the  three  sides  of  the  astro- 
nomical triangle,  and  putting 


° 


9o  -     =  a 

9o 

90 


We  find  that  the  solution  of  the  triangle  is  given  by 
Eqs.  6  and  7,  when  in  place  of  the  general  symbols  there 
employed  for  the  angles  of  the  triangle  we  substitute  the 
particular  values  which  they  have  in  this  case,  i.e., 


sin  5 


sin  (s  —  a)  sin  (s  —  b)  sin  (s— c)' 

cot  %aQ  =  k  sin  (s  —  d), 

cot  \t   =k  sin  (s-6),  (36) 

cot  Ag  =  k  sin  (s  —  c] . 


ROUGH  DETERMINATIONS    OF  TIME,  ETC.  65 

The  symbol  ao  here  introduced  to  represent  the  angle 
of  the  triangle  opposite  the  side  90°  —  d,  is  readily  seen 
from  Fig.  3  to  be  the  sun's  azimuth  reckoned  from 
north  toward  west,  and  if  we  desire  to  retain  the  cus- 
tomary mode  of  reckoning  azimuths,  from  the  south, 
we  must  write  A  =  i8o°  —  a0.  If  to  A  there  be  added 
the  observed  horizontal  angle,  H,  we  shall  of  course 
find  the  desired  azimuth  of  the  terrestrial  point,  viz., 

AM=i8o°  +  JJ-a0.  (37) 

The  angle  t  is  the  sun's  hour  angle  at  the  moment  of 
observation,  T,  and  we  have  therefore, 

T  +  4T=t  +  E  (38) 

from  which  to  determine  the  chronometer  correction. 
The  third  angle  of  the  triangle,  q,  is  introduced  only  for 
the  sake  of  the  following  control  which  it  furnishes  upon 
the  reduction.  Multiplying  together  the  last  three  equa- 
tions of  (36)  we  find, 

cot  \aQ  cot  \t  cot  \q  =  k  sin  s,  (39) 

a  relation  that  must  be  satisfied  within  one  or  two  units 
of  the  last  decimal  place  if  the  numerical  work  of  re- 
duction has  been  correctly  performed.  Another  useful 
"check"  thai  ih^uld  be  applied  midway  in  the  computa- 
tion is 

(5  -a)  +  (5  -  6)  +  (s  -  c)  =s.  (40) 

The  ambiguous  sign  of  k,  Eq.  36,  covers  the  two 
cases  of  observations  made  when  the  sun  is  west  or  east 


66 


FIELD    ASTRONOMY. 


of  the  meridian.     For  an  A.M.  observation  use  the  minus 
sign,  and  find  a0,  t  and  q,  all  negative  quantities. 

The  following  illustrative  record  and  reduction  relate 
to  observations  made  at  a  place  whose  latitude  and 
longitude  are  respectively  43°  4.6',  and  5h  58"*  west  of 
Greenwich. 

ALTITUDE  AND   AZIMUTH   OF  SUN'S  CENTRE. 

At  Station  A. — April  16,  1897. 
Engineer's  Transit,  B.     Watch  No.  6.     Observer,  C. 


Object. 

Circle. 

Watch. 

Vertical  Circle. 

Horizontal  Circle. 

Ver.  A. 

Ver.  B. 

Ver.  A. 

Ver.  B. 

Sta.  B 
Sun 
Sun 
Sta.  B 

R. 
R. 
L. 
L. 

h     m       s 

7  54  10 
259  14     o 
79  45     o 
187  54  10 

54  i° 
13  5° 
45  ° 
54  i° 

4    12    33 
4    15    28 

26  26     o 

25  57  20 

25     5° 

57      o 

REDUCTION. 


a 

10°  28'.  0 

sin  (s  —  a) 

9.4300 

do 

100° 

3°'  -4 

Vert.  Circ. 

26     11.5 

sin  (s  —  b) 

9-7157 

i8o°  +  jH" 

288 

24.7 

R'.  Table  IB 

-1.8 

sin  (s  —  c) 

9.8726 

AM 

187 

54-3 

h 

26      9.7 

Sum 

9.0183 

$ 

43       4-6 

sin  5 

9.9982 

t  (arc) 

63° 

50'.  o 

k2 

0.9799 

*  (time) 

1  5m  20* 

a 

79     32.o 

k 

0.4899 

E 

—  25 

b 

63     50-3 

t+E 

4 

T4     55 

c 

46     55-4 

cot  \a0 

9.9199 

T 

4 

J3     55 

25 

190     17.7 

cot  \t 

o.  2056 

jr 

+  1          0 

S 

95      8.8 

cot  \q 

0.3625 

s—a 

15    36-8 

Sum 

0.4880 

s-b 

£sin  s 

0.4881* 

5—C 

48     13.4 

Sum 

95      8.7* 

The  true  azimuth  of  Station  B  was  known  to  be 
187°  54'  10",  and  a  comparison  of  the  watch  with  standard 
time  furnished  as  the  true  value  of  AT,  +  57  seconds. 
The  differences  between  these  results  and  those  found 
in  the  preceding  solution  furnish  a  fair  idea  of  the  pre- 
cision to  be  expected  in  such  work. 


ROUGH  DETERMINATIONS  OF  TIME,  ETC.      67 

When  time  only  is  to  be  determined  from  an  observa- 
tion of  the  sun  the  measurement  of  the  angle  H  is 
unnecessary,  and  the  reduction  of  the  altitude  obser- 
vation may  be  made  by  the  method  of  §36.  Where 
azimuth  only  is  required  both  H  and  h  must  be  ob- 
served, and  the  sun's  azimuth  maybe  computed  some- 
what more  conveniently  than  above,  from  the  third  of 
Eq.  15,  written  in  the  form, 

sin  <£  sin  h  -  sin  $ 

Cos  A  =  -  — — —          -,  (41) 

cos  (j>  cos  h 

but  as  no  " check"  relation  accompanies  this  formula 
the  longer  computation  given  above  will  often  be  prefer- 
able. 

31.  Time  by  Meridian  Transits. — If  astronomical  ob- 
servations are  to  be  made  for  any  considerable  length 
of  time  at  a  given  station,  as  at  a  university,  it  will  be 
convenient  for  many  purposes  to  determine  the  azimuth 
of  a  permanently  marked  line,  at  one  end  of  which  an 
instrument  can  be  set  up  and  oriented.  If  a  theodolite 
be  thus  mounted  and  its  line  of  sight  brought  into  the 
meridian,  a  time  determination  may  be  very  simply  made 
by  observing  the  chronometer  time  of  transit  of  the 
sun's  preceding  and  following  limbs  past  the  vertical 
thread  of  the  instrument.  Since  the  thread  is  by  sup- 
position in  the  meridian,  the  hour  angle  of  the  sun  at  the 
mean  of  the  observed  times,  T,  is  zero  and  we  have 

JT  =  a  -T  (Sidereal), 
or  4T=E—T  (Mean  solar).  (42) 


68 


FIELD    ASTRONOMY. 


If  the  azimuth  of  the  line  is  well  determined,  this 
method  may  rank  as  an  approximate  rather  than  a 
rough  determination,  since  under  ordinary  circumstances 
there  must  be  an  error  of  nearly  2'  in  the  orientation 
of  the  instrument,  to  produce  an  error  of  6s  in  the  chro- 
nometer correction.  In  any  case  the  instrument  must 
be  carefully  levelled,  particularly  in  the  east  and  west 
direction,  and  in  the  following  example  the  readings 
of  the  striding  level  are  employed  as  a  control  upon  this 
adjustment. 

Observe  the  slight  variation  of  method  here  intro- 
duced in  order  to  obtain  in  place  of  a  single  observation 
two  observations,  one  Circle  R.,  and  one  Circle  L. 

TRANSITS    OF    SUN   FOR   TIME    DETERMINATION. 
At  Station  A.     April  17,  1897. 

Theodolite,  F.     Watch  No.  6.     Observer,  G. 
Instrument  oriented  on  Station  B. 


Circle. 

Ver.  A. 

Limb. 

Watch. 

Striding 
level. 

Reduction. 

0               / 

h.     m.     s. 

T  =  nh57m  37s 

R. 

359  30 

Pr. 

11   55   34 

1.4    12.6 

i2b+£  =  n   59     23 

L. 

180  30 

Fol. 

ii    59  40 

14.0      0.5 

jr=     +i   46 

By  a  comparison  with  standard  time  the  true  AT  referred  to  the 
local  meridian  was  found  to  be  +im  47*. 

The  telescope  of  an  engineer's  transit  is  usually 
capable  of  showing  a  first-magnitude  star  by  daylight 
whenever  the  sky  is  clear  and  blue,  and  such  a  star  is 
equally  available  with  the  sun  for  a  determination  of 
either  latitude  or  time.  In  the  spring  and  summer 
Sirius,  by  reason  of  its  great  brilliancy,  is  a  peculiarly 
favorable  object  for  such  observations  (see  Table  5 


ROUGH    DETERMINATIONS    OF    TIME,  ETC.  69 

for  the  approximate  right  ascension  and  decimation 
of  this  and  other  stars).  Even  the  brightest  of  these 
stars  is  not  a  conspicuous  object  by  daylight,  and  is  most 
readily  found  by  placing  the  telescope  in  the  meridian 
and  at  the  proper  zenith  distance,  z=  0— £,  and  await- 
ing its  arrival  in  the  field  of  view.  A  very  slight  error 
of  focus  in  the  telescope  will  render  the  star  invisible, 
and  this  adjustment  should  therefore  be  carefully  made 
upon  a  distant  terrestrial  object  before  setting  the  tele- 
scope for  the  star. 

32.  Orientation  by  Polaris. — If  a  rough  determination 
of  time,  latitude,  or  azimuth  is  to  be  made  by  night,  or 
if  a  theodolite  is  to  be  oriented  as  a  preparation  for  other 
work,  observations  of  the  Pole  Star  by  the  following 
method  will  be  found  especially  convenient,  since  no 
almanac  is  required  and  no  instrumental  equipment 
other  than  an  engineer's  transit  and  a  watch.  Even  the 
error  of  the  watch  need  not  be  known  in  advance.  If 
Polaris  were  exactly  at  the  pole  of  the  heavens  the 
instrument  might  be  oriented  by  pointing  directly  upon 
the  star  and  setting  the  verniers  of  the  horizontal  circle 
to  read  180°;  and  simultaneously  the  latitude  might  be 
determined  by  measuring  the  star's  altitude,  since  in  this 
case.  <p  =  h.  As  Polaris  is  actually  more  than  a  degree 
distant  from  the  pole  this  ideal  method  is  inapplicable, 
but  the  principles  upon  which  it  is  based  may  be  applied 
through  the  tables  at  the  end  of  the  book.  We  begin 
with  Table  4,  which  furnishes  the  amounts  a0,  b0,  by 
which  the  azimuth  and  altitude  of  Polaris,  as  seen  from 
a  given  place  at  a  given  time,  differ  from  the  corre- 


70  FIELD  ASTRONOMY. 

spending  coordinates  of  the  pole,  and  provision  is  made 
through  other  Tables,  i,  2,  3,  for  adapting  these  differ- 
ences of  azimuth  and  altitude  to  other  times  and  places. 
The  mathematical  basis  of  the  method  is  as  follows  : 

In  the  first  and  third  of  Eq.  14  let  us  replace  the 
declination,  d,  by  the  star's  polar  'distance,  p-=go°-d}  and 
for  the  ordinary  azimuth,  A,  let  us  substitute  an  azimuth, 
ao,  reckoned  from  the  north  point,  i.e.,  a=A  —  180°. 

When  the  resulting  equations, 

cos  h  sin  ao  =  —  sin  p  sin  t 

sin  h  =  sin  </)  cos  p  -f  cos  <j>  sin  p  cos  t,  (43) 

are  applied  to  Polaris  we  find  that  p  and  a0  are  small 
quantities  rarely  exceeding  i°or  2°,  whose  third  powers 
may  be  neglected  without  producing  error  exceeding 
i"  or  2",  and  to  this  degree  of  accuracy  we  may  use  in 
their  place  the  simpler  forms 

ao=  —  p  sec  h  sin  t 

cos  *-^2  sin2  J  tan  <£.  (44) 


The  quantities  given  in  Table  4  are  respectively  the 
terms  of  these  equations, 

a0=  —  p  sec  h  sin  t  bo=p  cos  t 

when  p  is  put  equal  to  70'.  5  and  h  is  equal  to  40°.  To 
adapt  these  tabular  quantities  to  other  values  of  p  and 
h  let  us  introduce  the  auxiliary  quantities, 

F=p  +  *]Qr.$  f=sech-r-  sec  40° 


ROUGH    DETERMINATIONS    OF    TIME,    ETC.  71 

and  substitute  for  the  star's  true  altitude,  h,  its  apparent 
altitude,  h'  ',  corrected  for  the  amount  of  the  refraction, 
R,  and  in  termi  of  thes3  find  from  Eq.  44, 


&'-<£  =    Fbo+R-^p2  sin2  t  tan  0.  (47) 

The  last  term  in  this  expression  is  very  small,  on 
account  of  the  factor  p2,  and  if  we  combine  its  average 
value  with  the  refraction,  R,  we  shall  find  that  the  result 
agrees  very  closely  with  the  combined  effect  of  refrac- 
tion and  the  sun's  parallax,  given  in  Table  IB,  under 
the  heading  R'. 

Eq.  47  may  be  used  to  compute  the  true  azimuth  of 
Polaris  and  the  difference  between  its  apparent  altitude 
and  that  of  the  pole,  hf—</).  For  this  purpose  we  need 
the  quantities  ao,  bo,  F}  f}  and  of  these  ao  and  60  are  to 
be  obtained  from  Table  4  with  the  star's  hour  angle,  t, 
as  the  argument.  The  value  of  t  at  any  moment  is 
readily  found  as  follows  :  Let  us  put 

M  =  Local  mean  solar  time  at  any  moment 
6=  Local  sideral  time  at  the  same  moment 
Q  =  Right  ascension  of  the  mean  sun  at  this  moment 
a  =  Right  ascension  of  Polaris, 
and  find  from  Eqs.  15  and  22, 


The  term  Q  —  a,  year  after  year  runs  through  nearly  the 
same  succession  of  values,  which  are  given  in  Table  3 


72  FIELD    ASTRONOMY. 

under  the  heading  D.  The  small  variation  of  these 
quantities  in  different  years  is  taken  into  account  by 
the  numbers  Y  of  Table  2 ,  and  replacing  Q  —  a.  by  its 
value  as  given  in  the  tables  we  find 

t  =  M  +  Y+D. 

The  number  Y  may  be  assumed  constant  throughout  a 
year,  while  D  is  to  be  interpolated  with  the  Greenwich 
mean  time,  to  the  nearest  tenth  of  a  day,  as  argument. 

Values  of  the  factors  /,  F  are  given  in  Tables  i  and  2 
with  the  argument,  the  altitude  of  Polaris,  and  the  year 
respectively.  The  value  of  F  may  usually  be  assumed 
constant  for  a  year  and  taken  from  Table  2  without 
interpolation ;  in  all  strictness,  however,  F  varies  slightly 
(aberration  +  precession)  and  Table  3  gives,  under  the 
heading  JF,  the  correction  required  to  reduce  the  value 
taken,  without  interpolation,  from  Table  2  to  its  true 
value  at  any  time.  Usually  these  corrections  to  F 
may  be  ignored,  and  Tables  i  and  2  are  then  most  con- 
veniently used  for  the  construction  of  a  supplementary 
table  like  the  following,  which  will  for  the  given  cal- 
endar year  save  all  further  reference  to  Tables  i  and  2 


1908.         (May). 

Y 


F 
R' 


i. oio 


h 

Ff 

42° 

1-037 

18 

43 

1-055 

44 

1.072 

Since  h  can  never  differ  from  the  latitude  by  much 
more  than  i°,  the  above  table  furnishes  all  values  of  the 


ROUGH   DETERMINATIONS   OF    TIME,    ETC.  73 

product  Ff  that  can  occur  in  latitude  43°  in  the  year 
1908,  and  this  product  when  required  should  be  ob- 
tained by  interpolation  from  such  a  table  rather  than 
by  multiplication  of  the  separate  factors,  F,  f.  Con- 
struct such  a  table  for  your  own  latitude  and  write  it 
in  pencil  at  the  bottom  of  Table  3.  The  formulae 
requisite  for  the  use  of  the  tables  may  now  be  collected 
and  put  in  the  form  shown  at  the  bottom  of  Table  4. 

From  these  equations  it  appears  that  the  tables 
furnish  all  \he  data  necessary  for  a  determination  of 
azimuth  or  latitude  from  observations  of  Polaris,  pro- 
vided the  local  mean  solar  time,  M,  is  known.  But  all 
too  often  the  relation  of  the  observer's  watch  or  chro- 
nometer to  local  time  is  very  ill  determined,  and  Table  5 
furnishes  a  method  for  supplying  this  defect  of  data. 
Its  several  columns  contain  the  name,  magnitude,  right 
ascension  and  declination  of  a  number  of  fast  moving 
stars,  which  may  be  observed  for  the  determination  of 
time.  For  the  explanation  of  most  of  these  quantities 
see  the  Introduction  to  the  Tables,  §  87;  but  we  may 
here  note  that  the  last  column  of  Table  5^  shows  under 
the  heading  MO  the  mean  solar  time  of  transit  of  each 
star  on  the  date  placed  opposite  it,  and  its  time  of 
transit  on  any  other  date  may  be  found  by  applying  to 
MQ  as  a  correction,  the  gain  of  sidereal  upon  mean  solaf 
time  during  the  interval  between  the  given  date  and  the 
date  shown  in  Table  5.  Take  the  amount  of  this  gain 
from  the  column  PP  for  D,  of  Table  3,  and  add  it  when 
the  given  date  is  earlier,  subtract  when  it  is  later  than 
the  date  given  in  Table  5 . 


74  FIELD    ASTRONOMY. 

The  values  of  M0  increase  slowly  with  the  lapse  of 
time,  the  average  rate  of  increase  being  a  minute  in 
fifty  years,  and  as  the  tabular  quantities  are  given  for 
the  epoch  1925  they  may  be  assumed  sufficiently  accu- 
rate during  the  first  half  of  the  twentieth  century.  The 
actual  time  of  a  star's  transit,  M^  is  subject,  however, 
to  considerable  changes,  depending  upon  the  observer's 
longitude  and  upon  the  influence  of  the  leap  year  cycle 
of  four  years.  The  combined  effect  of  these  influences 
is  represented  in  Table  Sc  ;from  which  may  be  taken, 
without  interpolation,  the  required  correction  to  adapt 
Mo  to  the  given  year  and  longitude.  Use  as  arguments 
the  observer's  longitude  from  Greenwich,  and  the  num- 
bers I,  II,  etc.,  indicating  that  the  year  for  which  Mo  is 
required  is  the  first  or  second,  etc.,  following  a  leap  year. 
The  presence  of  an  intercalated  day,  February  29,  in 
leap  year  requires  two  tables  for  such  a  year,  one  of 
which  is  to  be  used  for  the  months  of  January  and 
February,  the  other  during  the  remainder  of  the  year. 
See  the  example  §  33. 

The  use  of  these  tables  is  as  follows  :  After  the  instru- 
ment has  been  oriented  by  means  of  an  assumed  value 
of  the  watch  correction,  AT  ',  let  it  be  directed  to  the 
meridian  (south)  and  the  telescope  set  to  an  altitude 


/*=  (90°  - 


computed  for  that  one  of  the  stars  of  Table  5  that  is 
next  to  cross  the  meridian.  Observe  by  the  watch  the 
time,  r,  at  which  the  star  crosses  the  vertical  thread  of 


ROUGH    DETERMINATIONS    OF    TIME,    ETC. 


75 


the  instrument  and  note  whether  the  resulting  correction 
of  the  watch 


agrees  with  that  assumed  in  computing  the  orientation. 
If  the  difference  between  the  two  values  of  AT  does  not 
exceed  2m  it  may  usually  be  ignored  and  the  orienta- 
tion and  latitude  result  assumed  to  be  correct.  See  the 
following  example  for  a  case  in  which  the  difference  of 
the  JTs  is  too  great  to  be  neglected. 

33.  Illustration  of  the  Use  of  the  Tables.  —  An  observer 
who  supposes  his  latitude  to  be  about  43°  and  who  pos- 
sesses a  watch  approximately  set  to  Central  Standard 
Time  is  required  to  make  a  rough  determination  of 
azimuth  and  latitude  at  10  P.M.  on  May  19,  1908.  The 
relation  of  the  watch  to  local  time  is  quite  unknown, 
save  that  it  is  probably  a  few  minutes  slow,  and  assum- 
ing JT=  +  iom  there  is  made  the  following  preliminary 
computation  for  Polaris  and  Star  No.  29  of  Table  5A, 
using  the  auxiliary  quantities  given  on  p.  72,  which 
were  written  in  pencil  at  the  foot  of  Table  3.  The  sub- 
scripts i,  2  denote  quantities  that  refer  to  Polaris  and 
Star  No.  29  respectively. 


Preliminary  Computation. 
T+AT 

2   12 

13 

12  35 

-70' 

43   o 
41  50 
+  14 
1.04 
180  14 
June  21 


Final  Computation. 


Ff 


*  No.  29 

Af0  + Table  5? 

33  Days'  Change 

M2 

*> 


8 

2 

10 
20 


2 

10 
12 
46 


Obsd.  T2 

ioh     gr\ 

AT 

+  3 

T-i 

IO         0 

D+Y 

2       25 

t 

12       28 

Obsd.  /4 

4i°  55' 

Fbn 

—  i     ii 

R' 

+  r 

0 

43       5 

C0 

+  11 

Ff 

1.041 

180    ii 

Light-Polaris 

7    46 

Az.  of  Light 

187    57 

76 


FIELD    ASTRONOMY. 


RECORD   OF   THE  OBSERVATION. 
Tuesday,  May  19,  1908.     At  Station  Z. 


Object. 

Circle. 

Watch. 

Az.  Circle. 

Vert.  Circle. 

Polaris 

L. 

b.     m.     s. 
10     o     — 

0                / 

1  80      14 

0              / 

41       <?^ 

Lierht 

L. 

188       o 

O         2 

*No.  29    

L. 

10     9     21 

o       o 

2O      4'? 

A  few  minutes  before  10  o'clock  by  the  watch,  Polaris 
was  found,  using  the  computed  hi  to  facilitate  the  find- 
ing, the  horizontal  circle  was  set  to  read  the  computed 
azimuth,  AI,  and  at  ioh  om  by  the  watch  the  star  was 
brought  behind  the  intersection  of  the  cross  threads 
by  means  of  the  lower  motion,  without  altering  the 
reading,  A\.  The  time  and  circle  readings  were  re- 
corded in  the  first  line  of  the  record  as  given  above. 
The  reading  shown  in  the  second  line  was  then  taken 
to  a  distant  light  whose  azimuth  was  to  be  determined, 
and  this  reading  is  the  true  azimuth  of  the  light  pro- 
vided the  assumed  AT  is  correct.  To  guard  against 
error  in  this  assumption  the  star  TT  Hydrae,  No.  29,  of 
Table  5,  was  observed  as  set  forth  at  the  end  of  the 
preceding  section. 

The  difference  between  the  observed  time  and  the 
computed  M2  gives  +  3m  as  the  correction  of  the  watch 
relative  to  local  mean  solar  time,  and  since  the  differ- 
ence between  this  value  and  the  assumed  JT=  -fiom,  is 
too  great  to  be  neglected,  the  observations  were  recom- 
puted as  in  the  second  column ,  given  above  under  the 
heading,  Final  Computation. 


ROUGH    DETERMINATIONS   OF    TIME,  ETC.  77 

The  latitude  and  azimuth  there  obtained  do  not 
exhaust  the  data  furnished  by  the  observations,  and  we 
may  readily  find  a  more  accurate  value  of  AT  than  that 
above  derived.  Thus  finding  the  right  ascension  of 
TT  Hydras  from  Table  5  and  taking  Q  from  the  almanac, 
or  computing  it  by  §  19,  we  obtain  the  following: 


h  m  s 

14  i  9 

3  49  40 

10  ii  29 

10  9  33 


The  observed  time  of  transit,  T%,  has  here  been  in- 
creased by  1 2  seconds,  since  the  final  reduction  shows 
that  the  reading  o°  o'  at  which  the  instrument  was  set 
for  the  observation  of  n  Hydrae  does  not  correspond  to 
the  true  meridian,  but  is  3  minutes  east  of  it.  The 
average  southern  star  requires  about  4  seconds  of  time 
to  move  through  i'  of  azimuth,  and  the  corrected  time 
is  what  might  have  been  observed  had  the  instrument 
been  correctly  placed. 

33a.  Artificial  Illumination. — For  the  observation  of 
stars  by  night  there  must  be  provided  some  artificial 
illumination  for  the  telescope  as  well  as  for  the  verniers, 
since,  otherwise,  the  threads  that  determine  the  line  of 
sight  (cross- wires)  will  be  invisible.  For  this  purpose 
there  are  several  mechanical  devices  by  which  the  light 
from  a  bull's-eye  lantern  or  electric  hand-lamp  may  be 
reflected  into  the  field  of  view  of  the  telescope,  but  for 
a  small  instrument  none  of  these  possess  any  marked 


78  FIELD    ASTRONOMY. 

advantage  over  a  bit  of  candle-grease  dropped  in  the 
liquid  state  and  allowed  to  cool  upon  the  center  of  the 
objective.  Pare  it  down  thin  with  a  penknife  and  throw 
the  light  upon  it  along  a  line  but  little  inclined  to  the 
axis  of  the  telescope.  The  effect  of  the  grease  upon  the 
optical  performance  of  the  objective  is  quite  insensible. 


CHAPTER  VI. 
APPROXIMATE    DETERMINATIONS. 

34.  Latitude  by  Circum-meridian  Altitudes. — An  ob- 
vious method  of  refining  upon  the  rough  determination 
of  a  latitude  from  a  single  observation  of  the  meridian 
altitude  of  the  sun  or  a  star  (as  in  §  29)  is  to  measure  a 
series  of  altitudes  during  the  few  minutes  preceding  and 
following  the  maximum  h,  and  to  derive  from  all  these 
observations,  which  are  called  circum-meridian  altitudes, 
a  better  value  of  the  meridian  altitude  than  a  single 
measurement  can  be  expected  to  furnish.  Each  meas- 
ured altitude  will  usually  differ  from  the  maximum 
altitude  by  an  amount  called  the  reduction  to  the  merid- 
ian, and  this  reduction  may  be  accurately  computed 
if  either  the  hour  angle  or  the  azimuth  of  the  star  at  the 
time  of  observation  is  known. 

If  the  observations  are  made  with  a  sextant,  the  hour 
angle  will  be  most  convenient  for  the  reduction,  and 
the  time  of  each  observation  snouid  therefore  be  noted, 
to  the  nearest  second,  by  the  use  of  some  watch  or  other 
timepiece.  To  obtain  a  convenient  method  of  reduc- 
tion for  the  observations  we  put  t  =  o  in  the  equation, 

sin  h  =  sin  0  sin  d  +  cos  0  cos  d  cos  t,  (48) 

79 


80  FIELD  ASTRONOMY. 

and  obtain  for  the  maximum  altitude 

sin  h0  =  sin  0  sin  d  +  cos  0  cos  d.  (49) 

Since  in  the  cases  here  considered  the  hour  angles  are 
not  to  exceed  iom  or  15™,  we  may  put  cos/  =  i  —  ^2, 
and  subtracting  the  first  of  the  preceding  equations 
from  the  second  obtain, 


2  sin  %(h0  —  h)  cos  %(hQ  +  h)  =  J  cos  0  cos  d  .  t*t     (50) 
which  is  approximately  equivalent  to, 

.         COS  0  COS  d  t2 


This  is  the  equation  of  a  parabola  having  h0  for  its  maxi- 
mum ordinate  and  h  and  t  for  rectangular  coordinates; 
and  we  may  infer  from  it  that  if  the  sextant  readings 
be  plotted  as  ordinates  upon  cross-section  paper  with 
the  observed  times  as  abscissas,  the  resulting  curve  will 
be  a  parabola  whose  maximum  ordinate  will  be  the 
sextant  reading  corresponding  to  the  maximum  altitude 
of  the  body  observed. 

This  maximum  ordinate  may  be  read  directly  from 
the  curve,  or  it  may  be  derived  with  greater  precision 
by  means  of  the  theorem  that  the  area  included  between 
a  parabola  and  any  chord  perpendicular  to  its  axis, 
equals  two  thirds  of  the  length  of  this  chord  multiplied 
by  its  distance  from  the  vertex,  A  =  \xy.  The  inter- 
cept of  the  plotted  curve  upon  the  axis  of  x,  or  upon 
any  line  parallel  to  this  axis,  is  such  a  chord,  whose 


APPROXIMATE  DETERMINATIONS. 


81 


length  may  be  directly  measured,  and  the  distance  of  the 
vertex  from  this  axis  is  the  quantity  sought.  If,  there- 
fore, the  length  of  the  intercept,  x,  and  the  area  of  the 
corresponding  part  of  the  curve,  A,  be  directly  measured, 
we  have  at  once, 

3d 

J°        2   X' 

Friday  May,  4  1897. 

Sextant  No.  5096.     Index  corr.  —3'  34".     Observer,  C. 
Barometer  29.10.     Thermometer  69°  Fahr. 

Horizon  Roof  Direct.  Horizon  Roof  Reversed. 


(52) 


Limb. 

Watch. 

Sextant. 

h.     m.      s. 

0                 /                 // 

L. 

II    44    33 

125   36    10 

L. 

46    15 

125   39     o 

U. 

5°  27 

126  47  45 

U. 

5i   23 

126  48   20 

L. 

53   i5 

125  46   10 

Limb. 

Watch. 

Sextant. 

h.    m.      s. 

/       // 

U. 

II    55    45 

126  50     5 

U. 

58     3 

126    49    45 

U. 

12        0       8 

126  48      o 

L. 

5  43 

125    34  40 

L. 

6   30 

125    32    3° 

The  author  finds  from  the  area  of  the  curve  the  following  value  of 
from  which  the  latitude  is  derived  as  below. 


Index  corr. 

h' 
Ref.-Par. 


126°   1 8'  36' 

-3    34 

63       7     3i 

—  o     24 


63°    7'    7' 
26    52   53 
16    ii   41 
43      4   34 


The  axis  of  the  plotted  parabola  intersects  the  time  scale  at  nh 
55.0'",  and  comparing  this  number  with  the  local  mean  time  of  appar- 
ent noon,  i2h+£  =  nh  56m.6,  we  obtain  as  the  correction  required 
to  reduce  the  watch  to  local  mean  solar  time  Jr=+i.6m.  A 
value  of  AT  thus  determined  may  easily  be  in  error  by  20  or  30  seconds. 

Among  the  advantages  of  this  mode  of  treatment  of 
the  data  may  be  noted  that  each  observation  contributes 
its  appropriate  share  toward  determining  the  maximum 
altitude  of  the  body,  and  that  no  knowledge  of  the  error 
of  the  timepiece  is  required.  In  fact  the  correction  AT 


82  FIELD  ASTRONOMY. 

may  be  approximately  determined  from  the  curve,  by 
noting,  as  the  chronometer  time  of  apparent  noon,  the 
point  at  which  the  axis  of  the  parabola  intersects  the 
axis  of  x. 

Let  the  student  plot  the  preceding  observations  made 
upon  the  sun's  upper  and  lower  limb,  and  derive  from 
the  area  of  the  curve  the  sextant  reading  corresponding 
to  the  sun's  meridian  altitude.  Before  plotting,  each 
sextant  reading,  double  altitude,  must  be  corrected  by 
twice  the  sun's  semi-diameter,  interpolated  from  the 
almanac  for  the  date  of  observation,  i.e.,  ±31'  47",  in 
order  to  obtain  the  corresponding  reading  to  the  sun's 
center. 

For  an  approximate  method  of  determining  latitude 
from  altitudes  of  Polaris  the  student  may  consult  the 
American  Ephemeris,  Table  IV,  and  explanations  at 
the  end  of  the  volume. 

35.  Reduction  to  the  Meridian.  —  If  circum-meridian 
altitudes  are  to  be  measured  with  a  theodolite,  it  will 
usually  be  convenient  to  orient  the  instrument  and  deter- 
mine from  a  reading  of  the  horizontal  circle  the  azi- 
muth corresponding  to  each  observation.  A  graphical 
solution  may  then  be  made  precisely  as  in  the  case  of 
the  observed  times  treated  in  the  preceding  section,  or 
we  may  derive  from  Equations  15, 

sin  d  =  sin  0  sin  h  —  cos  0  cos  h  cos  A,  (53) 

and  from  this,  by  the  method  of  §  34,  we  find  the  relation, 

A2 
hQ— h  =  cos  0  cos  hQ  sec  d  .  —  +  etc.  (54) 


APPROXIMATE  DETERMINATIONS.  83 

Through  this  equation  and  the  known  values  of  A,  com- 
pute for  each  observed  altitude  its  own  reduction  to 
the  meridian. 

The  quantities  h0  —  h  and  A  are  here  supposed  to  be 
expressed  in  radians,  but  in  practice  it  is  convenient  to 
express  the  azimuth  in  minutes  and  the  reduction  to 
the  meridian  in  seconds  of  arc.  Representing  the  azimuth 
when  so  expressed  by  a',  we  make  in  Equation  54  the 
following  substitutions  : 

A  (radians)  -a'  .  -,  (*.-*)  (radians) 


and  uniting  into  one,  all  the  numerical  factors  that  are 
found  in  the  equation  as  thus  altered,  and  introducing 
the  symbol  /  as  an  abbreviation  for  the  product  of  all 
factors  not  containing  a'  ',  we  obtain, 

/  =  [7.9407]  cos  0  cos  hQ  sec  d, 

(*.-*)"  -/(oT- 

The  accents,  ',  "  ,  denote  that  the  marked  terms  are 
expressed  in  minutes  and  seconds  respectively.  Use 
an  estimated,  approximate,  value  of  h0  for  the  compu- 
tation of  /. 

The  preceding  results  cannot  be  directly  applied  to 
a  star  north  of  the  zenith,  since  for  such  a  star  the  azimuth, 
A,  is  a  large  quantity;  but  if  the  azimuth  be  reckoned 
from  the  north  point  instead  of  from  the  south,  i.e.,  if 
we  put  a0  =  i8o°  —  A,  we  may  derive  formulas  identical 
with  the  above,  which  therefore  apply  to  this  case  when 
a0  is  defined  as  the  supplement  of  the  azimuth.  For  a 


84  FIELD  ASTRONOMY. 

star  at  lower  culmination,  i.e.,  on  the  meridian  below 
the  pole,  the  altitude  is  a  minimum  instead  of  a  maxi- 
mum, and  the  reduction  to  the  meridian  must  therefore 
be  given  the  negative  sign.  Note  that  this  can  be 
accomplished  in  Equation  55  by  considering  d  to  repre- 
sent the  supplement  of  the  star's  declination  instead 
of  the  declination  itself.  These  formulae  for  reduction 
to  the  meridian  should  not  be  applied  in  the  case  of  stars 
whose  hour  angles  exceed  iom  or  i5m.  For  an  applica- 
tion of  the  formulae  see  §  73. 

36.  Time  from  Altitudes  near  the  Prime  Vertical.  — 
With  a  sextant  an  approximate  determination  of  time 
is  best  made  by  measuring  a  series  of  altitudes  of  the 
sun  or  a  star  when  the  body  is,  as  near  as  may  be,  due 
east  or  west,  noting  the  chronometer  time,  T,  of  each 
observation. 

The  formulae  for  the  transformation  of  coordinates 
furnish  for  each  such  observation  the  equation, 

sin  h  =  sin  0  sin  d  +  cos  0  cos  d  cos  t, 

which  is  readily  transformed  into, 

cos  t  =  sec  0  sec  d  sin  h  —  tan  0  tan  d,  (56) 

and  by  means  of  this  equation  the  hour  angle  corre- 
sponding to  each  observed  time  may  be  derived.  The 
chronometer  correction  will  then  be  furnished  by  one 
of  the  following  equations: 


For  the  Sun,  AT  =  E  +  t-T,  Local  Mean  Solar  Time. 
For  a  Star,    A  T  =  a  +  1  -  T,  Local  Sidereal  Time.          5  7 


APPROXIMATE  DETERMINATIONS. 


85 


The  symbol  E  denotes  the  equation  of  time.  Its  numeri- 
cal value  is  most  conveniently  derived  from  the  Solar 
Ephemeris,  p.  400  of  the  almanac. 

DOUBLE    ALTITUDES    OF    ARCTURUS,    NEAR    EASTERN 
PRIME    VERTICAL. 

Wednesday,  March  29,  1899. 

Sextant,  Cameron.     Chronometer,  B.     Observer,  C. 
Index  Corr.  +  18'  37".     Barom.  28.81.     Therm.  +19°  Fahr. 


Sextant.                 Chronometer. 

a  +  /.                                  A  T  • 

h 

.    m.    s. 

h.     m.     s.                              s. 

54   3°                 9 

30     5-5 

9  29     5-6              —59-9 

55      ° 

31   29 

30  27.9                  61.1 

55   3° 

32  5° 

31   5°-2                  59-8 

Horizon  roof  reversed. 

56     o 

34   i4 

33   12.5                  61.5 

56  3o 

35   36 

34  34-8                   61.2 

57     o 

36  58-5 

35   57-1                  61.4 

Mean  AT  —60  8 

0           /         // 

o      /       n      o       /       /r 

$ 

43     4  37 

Corr'd  Sext. 

54  48  37    57   18  37 

d 

sec  (f> 

19  42      9 

0.13642 

App't  h 
Refraction 

27   24   18 
1   55 

28  39   19 
i   49 

sec  6 

0.02620 

h 

27     22     23 

28  37   30 

tan  (f> 

9  .97082 

sin  h 

9.66255 

9  .68040 

tan  6 

9-55401 

sec  <f>  sec  8  sin  h 

9.82517 

9.84302 

sec  <j>  sec  d 

o.  16262 

tan  0  tan  6 

9-52483 

9-52483 

Subtract 

9  .99862 

0.03367 

h.     m.       s. 

cos  t 

9-52345 

9-55850 

a 

14   ii   6.1 

—  t 

4  42   0.5 

4  35      9-o 

a+t 

9  29  5-6 

9  35   57-i 

Adopted  JT=  -6oV8   (Local  sidereal). 

In  place  of  the  laborious  process  of  separately  reduc- 
ing each  observed  altitude  we  may  usually  treat  the 
mean  of  the  sextant  readings  and  the  mean  of  the  ob- 
served times  as  if  they  constituted  a  single  observation. 
When  the  observed  body  is  near  the  prime  vertical  the 
time  interval  covered  by  a  set  of  observations  which  it 
is  purposed  to  unite  into  a  mean  result  may  extend  to 
15  or  20  minutes  without  sensible  error,  but  the  error 


86  FIELD  ASTRONOMY. 

of  the  process  increases  rapidly  with  increasing  distance 
from  the  prime  vertical,  and  the  time  interval  must  be 
correspondingly  diminished. 

In  the  preceding  example  of  a  time  determination 
from  sextant  altitudes,  the  sextant  was  set  accurately  to 
a  set  of  readings  differing  by  a  uniform  interval  of  30', 
and  the  times  noted  at  which  the  observed  body  came 
to  the  corresponding  altitudes.  In  the  reduction  ad- 
vantage is  taken  of  this  circumstance  by  computing  the 
value  of  a  +  t  for  the  first  and  last  observations  only, 
and  interpolating  the  intermediate  values.  Observe 
that  the  columns  a  +  t  and  AT ,  although  placed  near  the 
beginning  of  the  reduction,  are  really  the  last  to  be  filled 
out. 

37.  Azimuth  Observations  at  Elongation.  —  An  excel- 
lent approximate  determination  of  the  azimuth  of  a 
terrestrial  mark  may  be  made  by  measuring,  with  a 
theodolite,  the  horizontal  angle  between  the  mark  and 
a  circumpolar  star  at  the  time  of  its  elongation,  i.e.,  its 
maximum  digression  from  the  meridian. 

It  may  be  seen  by  inspection  that  at  the  instant  of 
elongation  the  astronomical  triangle,  Fig.  3,  is  right- 
angled  at  the  star,  and  we  obtain  from  it, 

sin  A  e  =  cos  d  sec  0, 

(58) 
cos  te  =  cot  d  tan  0, 

where  the  subscript  e  shows  that  the  quantities  thus 
marked  relate  to  elongation  only,  and  Ae  is  measured 
from  the  north  toward  either  east  or  west  as  the  case 
may  require. 


APPROXIMATE  DETERMINATIONS.  87 

The  time  of  elongation  is  then  given  by 
Sidereal.  Qe=a±te 


~ 

ElonSatlon- 

Mean  solar. 


If  H  denote  the  measured  angle  between  the  star 
and  the  mark,  positive  when  the  mark  is  east  of  the 
star,  we  shall  have  as  the  azimuth  of  the  mark 

AM=iSo°+Ae+H.  (60) 

Equations  58,  59,  60  leave  nothing  to  be  desired  on 
the  score  of  simplicity  and  the  only  observation  required 
in  connection  with  them  is  the  measurement  of  the 
angle  H  at  the  time  Oe  or  Me.  This  is  frequently  done 
by  pointing  the  instrument  upon  a  star  a  little  before 
the  computed  time,  Me,  and  following  the  star's  motion 
in  azimuth  until  it  ceases  to  move  away  from  the 
meridian.  The  verniers  are  then  read  and  immediately 
thereafter  a  single  pointing  upon  the  azimuth  mark 
is  made.  This  method  is  convenient  when  the  time  is 
very  imperfectly  known  but,  in  general,  a  better  pro- 
cedure is  to  determine  the  time  to  the  nearest  minute, 
by  the  method  of  §  33,  or  otherwise,  and  to  make  at 
least  two  pointings  upon  both  mark  and  star,  one  Circle 
R  and  one  Circle  L,  to  eliminate  instrumental  errors, 
since  this  elimination  is  of  far  more  consequence  than 
any  precise  agreement  between  the  time  of  elongation 
and  the  actual  time  of  observation. 

Indeed  the  error  introduced  by  failure  to  measure 
the  angle  when  the  star  is  exactly  at  elongation,  may  be 
completely  remedied  as  follows  :  In  Fig.  6  let  SE'  represent 
the  vertical  circle  passing  through  a  circumpolar  star,  5, 


88 


FIELD  ASTRONOMY. 


which  is  T  minutes  away  from  its  elongation,  and  from  the 
pole,  P,  as  a  center  describe  a  circle  tangent  to  SE'.  A 
star  at  the  point  of  tangency,  Ef,  clearly  has  the  same 

azimuth  as  5  and  being 
at  the  elongation  in  its 
own  circle  this  azimuth 
may  be  found  through 
Eq.  58,  using  for  6  the 
declination  of  5  increased 
by  the  distance  between 
the  two  circles  of  the 
figure.  Calling  the  radii 
of  these  circles  p  and  pf 
respectively,  we  find  from 
the  spherical  triangle 
PSE't 


FIG.  6. 


tan  p'  =  tan  p  cos  T, 


which  may  be  transformed  by  substituting  for  cos  T  its 

T2    . 

value  i  —  —  into  the  relation 
2 


x  :  p  =  T2  :  n, 


(61) 


where  x  is  the  required  difference  of  radii  of  the  circles 
and  n  is  a  constant  number  whose  value  depends  upon 
the  units  employed  for  the  other  quantities.  When  p  is 
expressed  in  minutes  and  x  in  seconds  of  arc  we  find 

#  =  1745,  when  T  is  given  in  minutes  of  mean  solar  time, 
n  =  1755,  when  T  is  given  in  minutes  of  sidereal  time. 


APPROXIMATE  DETERMINATION. 


89 


Use  a  slide  rule  to  compute  the  value  of  #  from  Eq.  61 
and  find  the  azimuth  of  5"  from  the  equation, 

sin  ^4T=cos  (8+#)  sec  d>.  (62) 

Since  the  solution  here  given  is  only  approximate  it 
must  not  be  applied  to  a  star  too  far  away  from  its 
elongation  and  as  a  rough  guide  to  the  permissible  limit 
we  note  that  an  error  of  i"  in  x  need  not  be  feared 
when  the  product  tpx<  10,000  cot  <£,  T  being  expressed  in 
minutes  of  time,  p  in  degrees,  and  x  in  seconds  of  arc. 
Within  the  limit  thus  fixed  the  mean  of  several  consec- 
utive observations  of  a  star  may  be  reduced  as  a  single 
observation  by  substituting  in  place  of  T2  in  Eq.  61  the 
mean  of  the  T2s  corresponding  to  the  several  observations. 

The  following  illustrative  observations  at  elongation 
were  made  with  an  engineer's  transit,  using  the  method 
of  repetitions  (§  53),  two  pointings  in  the  set,  with  a 
reversal  of  the  instrument  between  them.  After  reversal 
the  bubble  of  the  azimuth  level  was  brought  back  to 
the  initial  position,  as  is  shown  by  the  level  readings, 
and  the  instrumental  errors  may  be  considered  as  well 
eliminated. 

Wednesday,  May  14,  1902. 
At  Station  A.     Inst.  No.  306.     Obsr.  C. 


Object. 

Circle. 

Point- 
ing. 

Watch. 

Horizontal  Circle. 

Levels. 

Ver.  A. 

Ver.  B. 

dUrs.  Min... 
Az.  mark  .  .  . 

L. 
R. 

I 
2 

8h  44m 
8     49 

4°    39'   5" 
251     24     o 

39'    5" 
24     o 

W.        E. 
9.8     12.2 
12.3     10.  o 

+  0.15 

90 


FIELD  ASTRONOMY. 


The  measured  angle  between  star  and  mark  is 

#  =  $(251°  24'  o"-4°  39'  5")  =  123°  22'   27".5, 
and    the    further  reduction    of    the    observation    is    as 
follows : 


t 

cotd 
tan<£ 
te 
a 

43°    4'  37"-° 
86    36    44  .8 
8.7723 
9.9708 

5h  47m  18" 
18      4      i 
3     27    36 
8    49      7 

*                   +   o  .8 

d+x 

86°  36'     45". 

6 

cos  (£+#) 

8.77148 

sec  (f> 

0.13642 

AT 

184      38    23    . 

2 

H 

123          22      27     . 

5 

AM 

308        o    51 

The  recorded  times  of  observations  are  from  a  watch 
approximately  regulated  to  Central  Standard  Time  and 
supposed  to  be  about  two  minutes  slower  than  local 
time.  Allowing  for  this  error  the  values  of  T  for  the  two 
pointings  are  —  2m  and  +3™  Respectively,  and  we  have 

therefore  x  \  2 03  =—    -'•:  1745  from  which  to  determine 

oo.  On  account  of  the  small  values  of  r,  tenths  of 
seconds  are  here  retained  in  the  computation  in  order 
to  illustrate  the  method,  but  the  result  for  AM  can  pre- 
tend to  no  such  degree  of  accuracy. 

A  frequent  source  of  substantial  error  in  similar 
determinations  lies  in  the  assumed  latitude,  and  unless 
this  is  well  known  the  elongation  method  should  be 
either  avoided  altogether  or  supplemented  by  observing 
the  elongation  of  another  star  on  the  opposite  side  of 
the  pole.  In  the  above  case  an  error  of  i'  in  the 
assumed  <j>  would  vitiate  AM  to  the  extent  of  4". 5,  but 
this  error  could  be  in  great  part  eliminated  by  observing 
also  the  western  elongation  of  51  H.  Cephei  which  occurs 


PLATE  II. 


An  American  Theodolite.     Diameter  of  Horizontal  Circle  8  inches. 
Approximate  Cost  $400. 

[To  face  p.  90.] 


APPROXIMATE   DETERMINATIONS.  91 

about  half  an  hour  after  that  of  d  Urs.  Min.  Adopt  the 
mean  of  the  resulting  values  of  AM.  It  is  always  ad- 
vantageous to  make  the  mean  of  the  pointings  coincide 
as  nearly,  as  may  be  with  the  computed  time  of  elonga- 
tion in  order  to  eliminate  any  effect  arising  from  error 
of  the  time  piece  employed. 

38.  Time  and  Azimuth  from  Two  Stars.  —  The  methods 
of  §  32  may  be  refined  so  as  to  furnish  the  required 
azimuth,  latitude  or  time  with  any  desired  degree  of 
accuracy,  but  such  refinement  naturally  is  at  the  cost 
of  increased  complexity  and  increased  labor  in  their 
application,  e.g.,  in  the  observations  we  may  so  use  the 
instrument,  Circle  R  and  Circle  L,  as  to  eliminate  its 
errors  of  collimation,  level,  etc.  ;  we  may  resort  to  the 
method  of  repetitions,  §  53,  to  secure  increased  accuracy 
in  the  horizontal  angle  between  mark  and  star,  etc. 
Corresponding  to  the  increased  precision  thus  obtained 
there  is  required  a  more  rigorous  method  of  reduction 
for  the  observed  data  and  the  mathematical  theory  of 
such  a  method  is  as  follows  : 

Let  M  denote  a  terrestrial  mark  between  which  and 
some  known  star  there  has  been  measured  the  horizontal 
angle,  H,  corresponding  to  the  time,  T,  as  indicated  by 
some  watch  or  other  time  piece,  and  let  h  denote  the 
altitude  of  this  star  measured  simultaneously  with  H. 
See  the  following  example  for  an  illustration  of  the 
method  of  observing  : 
Let  us  put 


The  Chronometer  Correction  (Sidereal) 
M  =Ao+y=True  azimuth  of  the  mark 


92  FIELD   ASTRONOMY. 

where  U  and  AQ  are  provisional  values  of  AT  and  AM 
obtained  from  the  orientation  method,  §  32,  or  other- 
wise, and  x,  y,  are  the  unknown  corrections  required  to 
transform  U  and  AQ  into  AT  and  AM.  To  determine 
the  values  of  oc  and  y  we  note  that  the  true  azimuth  of 
any  star  observed  may  be  expressed  in  the  form, 

A=AQ±H+y=Ai  +  y,  (63) 

where  AI  is  employed  as  an  abbreviation  for  the  meas- 
ured approximate  azimuth,  AQ±H.  Through  the  first 
of  Eq.  12  we  may  also  compute  the  star's  azimuth  A 
fxom  the  observed  time  T,  and  the  known  h,  d  and  a, 
thus 


sin  A  =cos  d  sec  h  sin  (T  +  x)  (64) 

where  r  =  T-f  £/—  a 

is  introduced  as  an  abbreviation  for  the  second  member 
of  the  above  equation.  Let  us  further  put  as  abbrevia- 
tions : 

sin  A'=cos  d  sec  h  sin  T 

C'=cos  d  sec  h  cos  r  (65) 

and  expanding  sin  (T+X)  in  Eq.  64  find 

sin  A  =  sin  A'  cos  x+  C'  sin  x.  (66) 


APPROXIMATE  DETERMINATIONS.  93 

When  %  and  sin  A'  are  small  quantities  whose  cubes  may 
be  neglected,  we  may  put  cos  #  =  i,  without  sensible 
error,  and  find  by  transposition  and  development  of 
sin  A  —  sin  A', 

2  sin  \(A-A')  cos  \(A+A')=C  sin  #.          (67) 

Again  neglecting  terms  of  the  order  #3  this  expression 
becomes 

A=A'+C'secA'-*,  (68) 

which  when  subtracted  from  Eq.  63  furnishes  the 
relation 

y-C'sec  A'-x-A'-Ai.  (69) 

The  quantities  y  and  A'  —  AI  of  this  equation  should 
be  expressed  in  seconds  of  arc,  but  the  chronometer 
correction,  x,  will  be  required  in  seconds  of  time,  and 
combining  with  sec  A'  the  numerical  factor  15  required 
for  this  purpose,  we  call  the  product  G  and  put  the 
coefficient  of  x  in  the  form  C'G.  Values  of  log  G  are 
given  in  the  following  short  table. 

THE  G  FACTOR. 

A'  logG  A' 

±0°  -1.1761+  i8o°io° 

1  .1762  I 

2  .1764  2 

3  -1767  3 

4  -1.1772+  4 

The  symbols  +  ,  —  placed  adjacent  to  log  G  indicate 
the  essential  sign  of  G,  +  for  a  northern,  and  —  for  a 


94  FIELD  ASTRONOMY. 

southern  star,  and  they  must  be  heeded  in  computing 
the  coefficient  of  x. 

Every  observed  star  will  furnish  an  equation  of 
the  type  (Eq.  69),  and  two  stars  will  therefore  suffice 
for  the  determination  of  both  y  and  x.  It  is  a  matter 
of  consequence,  however,  to  choose  wisely  the  particular 
stars  that  are  to  be  observed.  One  of  them  should 
always  be  near  the  pole  and  Polaris  should  usually  be 
chosen,  wherever  it  may  be  in  its  diurnal  path.  The 
other  star  should  be  on  the  opposite  side  of  the  zenith 
from  Polaris,  and  because  of  the  assumptions  made  in 
Eq.  66  it  should  be  observed  only  when  near  the  merid- 
ian ;  e.g.,  when  #  is  as  great  as  3™  the  star's  azimuth 
should  not  exceed  ±3°. 

We  now  collect  and  arrange  our  formulae  as  follows  : 


r=T+U-a 

C=G  sec  h  cos  d  cos  T 
sin  A'=sec  h  cos  d  sin  T 
y+Cx=A'—Ai     (2  Equations) 


4T=U+x. 

Example.  —  The  application  of  the  method  is  illus- 
trated by  the  following  record  and  reduction  of  obser- 
vations made  with  the  instrument  shown  in  Plate  I,  and 
an  ordinary  watch. 


APPROXIMATE   DETERMINATIONS. 


95 


Tuesday,  April  23,  1907. 
At  Station  E.     Azimuth  of  Asylum  Light. 
Observer.  C. 


Object. 

Circle. 

Watch. 

Horizontal  Circle. 

Vertical  Circle. 

Levels. 

Ver.  A 

Ver.B 

Ver.  I. 

Ver.  II. 

Light  

R. 
R. 
R. 
L. 
L. 
L. 

h  m      s 
90     — 

2    16.5 

5  ii 
10  14 
14     8.5 
9  17    — 

187   54     o 
358   14  15 
i79     5  40 
359    8  30 
181  41  20 
7  54  3° 

53  45 
14  15 
5  3° 
«  25 
40  45 
54  10 

Azimuth  Level 
W.         E. 

25-7       9-3 
9-i     25.6 

d  Crateris.  . 
Polaris.  .  .  . 
Polaris  
d  Crateris.. 
Light  

32  40  45 
42     9  30 
42     3  15 
32  38  45 

39  IS 
7     o 
3  45 

39     o 

+  0.15 
Altitude  Level 
N.          S. 
2.O         8.O 
8.0         2.2 

—  O.I 

REDUCTION. 


Star. 

Polaris. 

5  Crateris. 

U 

+  2h  6m  Os 

at 

xh  241*1  46s 

nh  14111  428.3 

X 

+  25.1 

T 

9      7     42 

9      8     12  .5 

AT 

2     6  25.1 

T+U 

ii    13     42 

II      14       12    .5 

Q 

2     4     0.3 

T 

9    48     56 

—  o      o     29  .8 

AT 

+  2    24-8 

T  (arc) 

147°  14'      o" 

-  o°    7'    27" 

8 

88    48      31 

—  14    16     41 

J 

4i 

179     7        i 

359    57     39 

A, 

187°  54'  6" 

h 

42     6 

32    39 

y 

+  59 

G 

i  .  1762 

i  .  17617* 

A* 

187    55  5 

COST 

9-9247w 

o  .0000 

sec  h 

o.  1296 

0.0747 

cos  S 
sin? 

8.3179 
9-7334 

9.9864 
7-3359" 

Equations. 

C 

9.5484** 

I.2372W 

y-  0.35*=+  50" 

sin  A' 

8.1809 

7-397CW 

y-i7.27*=-374 

A' 

179°  7'  5i" 

359°  5i'  25" 

The  instrument  was  approximately  oriented  before 
beginning  the  observations,  a  precaution  that  it  is  always 
well  to  take,  since  we  may  then  assume  as  provisional 


96  FIELD  ASTRONOMY. 

azimuths,  A0,  A\,  the  mean  readings  of  the  horizontal 
circle  corresponding  to  mark  and  star  respectively.  Sim- 
ilarly, taking  the  mean  of  the  observed  times  for  each 
star  and  comparing  the  time  for  d  Crateris  with  the 
right  ascension  of  that  star,,  we  find  with  sufficient 
approximation,  £7  =  +2h6m.  The  instrument  was  re- 
levelled  when  reversed  by  bringing  the. bubble  back  to 
the  position  it  occupied  at  gh  sm  and  the  bubble  readings 
in  the  last  column  show  that  in  the  mean  the  errors  of 
level  are  so  small  that  their  effect  may  be  ignored.  See 
§§42  and  50. 

The  two  values  of.  AT  above  derived  relate  respect- 
ively to  sidereal  and  to  mean  solar  time.  A  com- 
parison of  the  watch  with  the  standard  clock  of 
the  Washburn  Observatory  made  immediately  after 
this  observation  gave  as  its  true  correction,  AT  = 

+  2m24s.6. 

In  the  reduction,  the  coordinates  of  the  stars  and 
the  quantity  Q  have  been  taken  from  the  almanac.  For 
comparison  let  the  student  derive  their  values  from 
the  tables  at  the  end  of  this  book  and  Eq.  19. 

The  altitudes,  h,  that  appear  in  these  formulae  need 
not  be  observed  with  great  precision,  an  error  of  i'  or 
even  2'  having  in  general  no  appreciable  effect  upon 
the  values  of  x  and  y.  Indeed  the  observations  of  alti- 
tude may  be  entirely  omitted  and  adequate  values  of  h 
computed  from  the  orientation  tables  for  Polaris,  and 
from  the  declinations  and  hour  angles  for  the  southern 
star.  If,  however,  these  altitudes  are  accurately  ob- 
served their  values,  corrected  for  refraction  and  in- 


APPROXIMATE  DETERMINATIONS.  97 

strumental  error,  may  be  made  to  yield  an  excellent 
determination  of  the  latitude.  We  may  use  for  their 
reduction,  in  the  case  of  the  southern  star,  the  methods 
°f  §§  34,  35  and  for  Polaris  the  following  equivalent  of 
Eq.  44. 

6=h~-p  cos  (r+x)+C" 

C"  =  (4(180 -AS))*  sin  2h. 

In  the  last  equation  the  value  of  AI  must  be  ex- 
pressed in  degrees  and  the  resulting  value  of  C"  will 
then  be  given  in  seconds  of  arc,  e.g.,  if  ^1  =  178°  48' 
h  =  45°,  we  find 


=  4X1-2    sin9o   =  23. 

It  may  be  noted  that  when  the  chronometer  correc- 
tion, AT,  is  sufficiently  well  known  we  may  put  x=o, 
omit  the  observation  of  the  southern  star  and  find 
AM  from  the  equation  furnished  by  the  Polaris  obser- 
vation alone.  Similarly,  if  the  azimuth  of  the  mark  is 
given,  we  may  put  y=  o,  omit  the  observation  of  Polaris 
and  determine  AT  from  an  observation  of  a  southern 
star,  or,  by  daylight,  from  an  observation  of  the  sun 
when  near  the  meridian.  Compare  with  the  method 
of  §  31.  When  applied  to  a  star  this  procedure  will 
often  be  of  advantage  in  connection  with  precise  deter- 
minations of  azimuth. 

Errors  of  Levelling. — A  major  source  of  error  in  results 
obtained  as  above  is  found  in  imperfect  levelling  of  the 
instrument,  and  it  should  be  avoided  by  reading  the 
azimuth  level  (§  50)  at  some  time  during  the  first  half 


ASTRONOMY. 

of  the  set  of  observations  and  relevelling  the  instru- 
ment after  reversal.  Bring  the  .bubble  back  to  the 
exact  place  that  it  occupied  in  the  tube  when  first 
read  and  record  the  second  readings.  Whatever 
error  affected  the  first  half  of  the  set  will  now  be 
balanced  by  an  equal  and  opposite  error  in  the  second 
half. 

It  may  happen  that  the  recorded  level  readings, 
before  and  after  reversal,  do  not  agree,  and  in  this  case 
the  angle,  b',  that  the  vertical  axis  makes  with  the 
plane  of  the  meridian  may  be  found  from  the  recorded 
level  readings  (§§  42,  43)  and  a  corresponding  correction 
applied  to  the  time  and  azimuth  found  through  the 
vitiated  values  of  x  and  y.  From  the  spherical  triangle 
formed  by  the  pole,  the  zenith  and  the  point  of  the  sky 
towards  which  the  vertical  axis  is  directed  we  find, 

Ax=  +b'  sec  <£  Ay=  +  6'  tan  <£, 

which  are  the  required  corrections  for  level  error. 
Count  &  as  positive  when,  on  the  whole,  the  east  end 
of  the  azimuth  level  is  higher  than  the  west  end.  As- 
suming the  value  of  a  level  division  to  be  15"  (d=f. 5) 
we  find  from  the  observations  on  p.  95,  b'=  +i", 
Ax=  + 1 ".3  =  +os.i,  Jy=  + 1",  which  are  quantities  quite 
inappreciable  for  the  present  purpose. 


STUDENTS' 
OBSERVATORY 


INSTRUMENTS. 

IN  the  several  determinations  thus  far  considered 
we  have  for  the  most  part  assumed  that  the  data  fur- 
nished by  the  instruments  employed  were  free  from 
purely  instrumental  errors,  and  in  approximate  work 
this  may  usually  be  done  if  due  care  has  been  bestowed 
upon  the  adjustments.  But  where  a  higher  degree  of 
precision  is  required  it  becomes  necessary  tb  study  the 
instrument  employed,  as  being  in  itself  a  source  of  errors 
that  need  to  be  eliminated,  and  we  must  turn  therefore 
to  a  more  detailed  consideration  of  some  of  the  instru- 
ments used  in  field  astronomy  before  taking  up  the  class 
of  methods  called  accurate. 

42.  The  Spirit-level.  —  The  spirit-level  is  used  in 
astronomical  practice  to  measure  small  deviations  of  a 
line  or  surface  from  a  vertical  or  horizontal  position, 
and  incidentally  to  adjust  a  part  of  an  instrument  to 
such  a  position.  It  consists  essentially  of  a  glass  tube 
bent  or  ground  into  an  arc  of  a  circle  of  large  radius  and 
so  mounted  that  the  plane  of  this  circle  is  approximately 
vertical.  The  tube  being  nearly  filled  with  ether  and 

its  ends  hermetically  sealed,  the  small -volume  of  air  or 

99 


100  FIELD  ASTRONOMY. 

vapor  that  remains  in  the  tube  is  collected  into  a  bubble 
which  always  stands  at  the  highest  point  of  the  circle, 
so  that  a  line  drawn  from  its  middle  point  through  the  cen- 
tre of  curvature  of  the  tube  is  vertical.  The  upper 
surface  of  the  tube  is  usually  provided  with  a  scale  of 
equal  parts,  and  the  position  of  the  bubble  in  the  tube 
is  determined  by  the  readings  of  its  ends  upon  this  scale. 
The  angle  subtended  at  the  centre  of  curvature  of  the 
tube  by  the  space  between  two  consecutive  lines  is  called 
the  value  of  a  division  of  the  level,  and  this  value,  which 
will  be  represented  by  2d,  is  required  for  transforming 
the  indications  of  the  level  into  seconds  of  arc.  Note 
that  d  represents  one  half  the  value  of  a  level  division. 

Let  such  a  level  be  supposed  attached  to  a  theodolite, 
the  inclination  of  whose  vertical  axis  to  the  true  vertical 
is  to  be  determined  from  readings  of  the  bubble.  We 
are  here  concerned  with  angular  measurements,  e.g.,  the 
angle  that  the  axis  makes  with  the  true  vertical;  the 
angle  moved  over  by  the  level  bubble,  as  seen  from  the 
centre  of  curvature  of  the  tube,  when  the  instrument 
is  turned  from  one  position  to  another;  etc.,  and  as  the 
simplest  method  of  dealing  with  these  angles  we  shall 
imagine  the  whole  apparatus  projected  radially  upon 
the  celestial  sphere,  so  that  the  arc  joining  the  points 
in  which  any  two  projected  lines  meet  the  sphere,  meas- 
ures the  angle  between  these  lines.  This  method  of 
analysis  by  projecting  the  parts  of  an  instrument  upon 
the  sphere  is  in  common  use,  and  the  student  should 
acquire  a  clear  conception  of  the  simple  case  to  which 
it  is  here  first  applied. 


INSTRUMENTS.  101 

To  determine  the  relation  of  the  bubble  readings  to 
the  required  inclination,  we  imagine  the  axis  of  the  in- 
strument and  the  plane  of  the  level  extended  until  they 
meet  the  celestial  sphere,  as  in  Fig.  7,  which  represents 


L 

FlG.  7.— Theory  of  the  Spirit-level. 

a  small  part  of  the  sphere  adjacent  to  the  zenith,  Z. 
In  this  figure  V  is  the  point  in  which  the  produced  axis 
meets  the  sphere,  and  LB  is  the  trace  of  the  plane  of  the 
level  tube  upon  the  sphere.  The  projection  of;  thfe,  n^i^ 
die  of  the  bubble  upon  the  sphere  must  be  at^  the,  point, 
in  LB  nearest  to  Z,  and  found,  therefore,  by  ^  letting  iall 
a  perpendicular  from  Z  upon  LB.  If,  now,  the  theod- 
olite be  turned  180°  in  azimuth,  i.e.,  rotated  about  V 
as  a  pivot,  the  level  tube  will  be  revolved  about  V  as  a 
centre,  into  the  position  L'B' ',  and  the  point  B  will  fall 
at  B',  but  the  middle  of  the  bubble  will  stand  at  B" 
instead  of  B',  since  this  is  now  the  point  nearest  to  the 
zenith.  From  elementary  geometrical  considerations, 
VM  =  %B'B",  where  B'B"  is  the  space  moved  over  by 
the  level  bubble  when  the  instrument  is  turned  from 


102  FIELD  ASTRONOMY. 

one  position  to  the  other,  and  VM  is  the  projection 
upon  the  plane  of  the  level  tube  of  the  arc,  VZ,  that 
measures  the  angle  between  the  axis  of  rotation  and 
the  true  vertical.  Calling  this  projection  b  and  repre- 
senting by  a',  b',  a"  ,  b"  the  scale  readings  of  the  ends 
of  the  bubble  in  the  two  positions,  we  have 


(  , 


It  is  customary  to  record  the  several  readings  in  the 
form, 

(Symbols  used  above.)       (Actual  observations.) 

N          S  N  S 

a1        b'  l6-4        39-6 

a"       b"  32.2  9.1 

7-35 

The  letters  N  and  S  denote  the  north  and  south  ends 
of  the  level  tube,  or  some  equivalent  system  of  distinguish- 
ing between  them. 

•43.  Discussion  of   the  Level   Readings.  —  The    student 
Should  no\Y  note  that : 

;  ^K^f1^ ,  coefficient  of  d  in  Equation  79  is  the  mean  of 
the  diagonal  differences  in  the  square  array  formed  by 
the  four  numbers  tabulated  in  the  preceding  example. 
This  example  represents  the  manner  in  which  level  read- 
ings should  be  recorded,  and  the  mean  of  the  diagonal 
differences,  7.35,  written  below  the  line,  should  be  worked 
out  and  entered  with  the  record. 

(b)  If  the  bubble  readings  have  been  correctly  taken 
and  there  is  no  change  in  the  length  of  the  bubble  during 
the  observation,  these  differences  must  be  equal,  one 


INSTRUMENTS.  103 

to  the  other,  thus  furnishing  a  check  upon  the  accuracy 
of  the  level  readings,  which  should  always  be  applied 
immediately  after  recording  them.  If  the  temperature 
is  changing  rapidly,  the  length  of  the  bubble  may  be 
changed  and  the  check  impaired  without  necessarily 
diminishing  the  accuracy  with  which  b  is  determined. 

(c)  If  the  greatest  of  the  four  numbers  stands  in  the 
column  marked  N,  the  north  end  of  the  level  tube  is  on 
the  whole  higher  than  the  south  end,  and  the  vertical 
axis  is  tipped  toward  the  south.     Determine  the  sign 
of  b  in  this  manner. 

(d)  The  zero  of  a  level  graduation  is  sometimes  placed 
at  one  end  of  the  scale  and  sometimes  in  the  middle,  but 
the  method  of  record  and  reduction  given  above  applies 
to  both  cases. 

(e)  It  is  apparent  from  the  figure  that  the  point  of 
the  level  tube  midway  between  Bf  and  B"  marks  that 
radius  of  the  level  tube  which  is  most  nearly  parallel 
with  the  rotation  axis  of  the  instrument.      Since  this 
radius  ought  to  pass  through  the  middle  point  of  the 
scale,  and  does  so  pass  when  the  level  is  in  adjustment, 
we  have  as  the  error  of  adjustment  of  a  level  numbered 
continuously  from  one  end  to  the  other, 


(80) 


where  s  represents  the  total  number  of  divisions  in  the 
level  scale.  In  the  example  given  above  5  =  50  and 
e  =0.7  division. 

The  essential  element  in  the  determination  of  b  is 


104  FIELD  ASTRONOMY. 

the  reversal  of  the  level  with  the  resulting  displace- 
ment of  the  bubble,  and  it  is  a  matter  of  indifference 
whether  this  displacement  is  produced  by  revolving 
the  level  about  a  vertical  axis  to  which  it  is  attached, 
as  in  the  case  considered  above,  or  by  picking  the  level 
up  bodily  from  a  plane  or  line  upon  which  it  stands, 
turning  it  end  for  end  and  replacing  it  in  the  reversed 
position,  as  is  done  in  measuring  the  inclination  of  an 
approximately  horizontal  axis.  Let  the  student  show 
that  the  inclination  of  this  axis  to  the  plane  of  the 
horizon  may  be  obtained  from  the  bubble  readings 
exactly  as  the  inclination  of  the  vertical  axis  was  deter- 
mined above.  The  greatest  of  the  four  readings  is 
adjacent  to  the  high  end  of  the  axis.  Determine  in  this 
way  the  inclination  of  the  horizontal  axis  of  a  theodolite. 
A  fine  level  is  an  exceedingly  sensitive  instrument 
and  requires  great  care  in  its  use.  Unless  unusually  well 
supported  its  readings  may  be  vitiated  by  the  observer 
passing  from  one  side  of  it  to  the  other,  or  even  by  shift- 
ing his  weight  from  one  foot  to  the  other.  Therefore 
observe  the  following  precepts: 

1 .  Keep  away  from  the  level  as  much  as  possible. 

2.  Don't  allow  the  sun  to  shine  upon  it. 

3.  Don't  hold  a  source  of  heat,  e.g.,  a  lamp  or  your 
own  hand,  near  a  level  longer  than  is  strictly  necessary. 

4.  If  the  level  has  a  chamber  with  reserve  supply  of 
air  at  one  end  of  the  tube,  use  it  to  regulate  the  length 
of  the  bubble,  keeping  this  always  about  one  half  as 
long  as  the  scale. 

5.  Make  the  inclinations  that  are  to  be  measured 


INSTRUMENTS.  105 

as  small  as  possible,  in  order  to  avoid  any  considerable 
run  of  the  bubble  and  the  resulting  effect  of  possible 
irregularities  in  the  level  tube. 

44.  Value  of  a  Level  Division. — The  value  of  a  level 
division  is  most  conveniently  determined  by  measuring 
with  a  micrometer,  or  finely  graduated  circle,  the  vertical 
angle  through  which  its  tube  must  be  tipped  in  order 
to  cause  the  bubble  to  run  past  a  given  number  of  divi- 
sions of  the  scale.  If  the  necessary  apparatus  for  such 
a  determination  is  not  at  hand,  the  following  method 
will  furnish  equally  good  results  and  requires  only  an 
engineer's  transit,  to  which  the  level  must  be  attached 
with  its  plane  approximately  vertical. 

Let  the  instrument  be  firmly  set  up  but  very  much 
out  of  level,  e.g.,  with  its  vertical  axis  making  an  angle, 
f,  with  the  true  vertical  amounting  to  2°,  more  or  less. 
See  p.  109  for  a  method  of  determining  the  exact  value 
of  this  angle,  which  will  be  required  in  the  reduction  of 
the  observations.  If  the  transit  is  now  turned  slowly 
about  its  vertical  axis  (azimuth  motion),  the  level -bubble 
will  run  back  and  forth  in  its  tube,  and  two  positions  of 
the  instrument  may  be  found  at  which  the  bubble  will 
come  to  the  middle  of  its  scale.  We  shall  designate  the 
readings  of  the  azimuth  circle  corresponding  to  these  two 
positions  by  Al  and  A2. 

Any  slight  turning  of  the  instrument  either  way 
from  A  j^  or  A2  will  cause  a  corresponding  slight  motion 
of  the  bubble,  and  to  determine  the  relation  of  the  bubble 
readings  to  the  corresponding  circle  readings  we  resort 
to  Fig.  8,  which  represents  a  portion  of  the  celestial 


106  FIELD  ASTRONOMY. 

sphere  adjacent  to  the  zenith,  Z.  V  is  the  point  in 
which  the  deflected  axis  of  the  instrument  meets  the 
sphere,  and  SV  is  the  trace  upon  the  sphere  of  the  plane 
of  the  level-tube,  which  is  assumed  to  have  been  adjusted 
approximately  parallel  to  the  vertical  axis  of  the  transit. 
Small  errors  in  this  adjustment  are  of  no  consequence. 


FIG.  8.—  Determination  of  d. 

As  the  instrument  is  turned  in  azimuth,  carrying  the 
level-tube  with  it,  the  arc  SV  must  revolve  about  V  as 
a  pivot,  and  the  amount  of  its  rotation  will  be  measured 
by  the  successive  readings  of  the  azimuth  circle.  It 
may  be  seen  readily  that  the  angle  T  of  the  figure  corre- 
sponding to  any  particular  circle  reading,  A,  is  given 
by  the  equation 

A,)-^,  (82) 


t-f  A2)  being  the  circle  reading  at  which  SV  coincides 
with  VZ. 


INSTRUMENTS.  107 

Since  a  level-bubble  always  stands  at  the  highest 
point  of  its  tube,  the  point  nearest  the  zenith,  we  may 
find  the  point  in  the  figure  corresponding  to  the  middle 
of  the  level -bubble  by  drawing  from  Z  an  arc  of  a  great 
circle  perpendicular  to  5 V,  and  the  intersection,  5,  will 
be  the  required  point.  In  the  right-angled  spherical 
triangle  SVZ  thus  formed  we  have  the  relation, 

tan  p  =  tan  ?  cos  T,  (83) 

in  which  p  measures  the  distance  of  the  middle  of  the 
bubble  from  the  fixed  point  V.  To  find  the  effect  upon 
p  of  any  small  variation  in  r,  i.e.,  to  find  how  far  the 
bubble  will  run  when  the  instrument  is  turned  slightly 
in  azimuth,  we  differentiate  this  equation  and  obtain 

—  dp=  tan  f  cos2  p  sin  r  dr,  (84) 

and  substituting  in  place  of  these  differentials,  small 
finite  increments  of  the  respective  quantities,  we  obtain 

2d(b'-b")  =tan  r  cos2  p  sin  r  (A' -A"),         (85) 

where  d  represents  the  value  of  half  a  level  division  and 
b'  and  b"  are  the  scale  readings  of  the  middle  of  the  bubble, 
corresponding  to  the  circle  readings  A'  and  A" . 

Equation  85  may  be  used  to  determine  the  value  of  d, 
but  whenever  ordinal  y  care  is  bestowed  upon  the  ad- 
justment of  the  level,  i.e.,  to  make  the  radius  passing 
through  the  middle  point  of  the  scale  parallel  to  the  ver- 
tical axis  of  the  theodolite  (Equation  80),  the  readings 
Aj  and  A  2  will  be  so  nearly  180°  apart  that  we  may  put 
7  =  90°,  cos£  =  i,  for  all  positions  of  the  bubble  within 


108  FIELD  ASTRONUMY. 

the  limits  of  its  scale,  and  thus  obtain  in  place  of  Equa- 
tion 85  the  simpler  relation 

~ 


In  this  equation  A'  —  A"  and  b'  —  b"  are  to  be  derived 
from  the  readings  of  the  horizontal  circle  and  level, 
respectively,  and  in  making  observations  for  their  deter- 
mination it  is  well  to  bring  the  bubble  as  near  as  may 
be  to  one  end  of  the  tube  and  set  the  circle  to  read  the 
nearest  integral  10'.  Then  turn  the  instrument  to  each 
successive  10'  or  20'  reading,  and  record  the  readings 
of  the  bubble  until  the  former  has  traversed  the  entire 
length  of  its  scale,  after  which  repeat  the  operation  in 
the  inverse  order,  using  the  same  circle  settings  as  before. 
With  reference  to  the  direction  of  the  bubble's  motion 
these  two  series  will  be  designated  as  Forward  and 
Backward.  Having  completed  these  observations,  turn 
the  instrument  to  the  second  position  in  which  the  bubble 
plays,  e.g.,  from  A^  to  A2,  and  make  a  similar  double  set 
of  readings. 

The  readings  obtained  at  any  two  settings  of  the 
instrument  will  determine  values  of  A'  —  A"  and  b'  —  b"  , 
and  therefore  a  value  of  d,  but  it  is  advisable  to  secure 
a  considerable  number  of  these  determinations,  ranging 
over  the  whole  length  of  the  level  -tube,  in  order  to  test 
its  uniformity.  Supposing  such  a  series  to  have  been 
made,  the  manner  of  forming  the  differences  b'  —  b" 
illustrated  below,  may  be  followed  with  advantage,  i.e., 
subtract  the  first  b  from  the  first  one  following  the 


INSTRUMENTS.  109 

middle  of  the  set,  the  second  b  from  the  second  one  after 
the  middle,  etc. 

The  angle  7-  of  Equation  86  should  be  determined 
at  the  time  of  deflecting  the  axis,  as  follows :  After  having 
carefully  levelled  the  instrument,  take  a  reading  of  the 
vertical  circle  when  the  line  of  sight  is  directed  toward 
a  fixed  mark,  that  we  may  call  P.  By  means  of  the 
levelling  screws  deflect  the  axis  exactly  toward  or  from 
P  through  some  convenient  angle,  e.g.,  i°  if  the  vertical 
circle  reads  to  seconds,  3°  if  it  reads  only  to  minutes,  and 
again  point  upon  P  and  read  the  circle.  The  difference 
of  the  two  readings  is  the  value  of  ?.  To  make  sure  that 
the  deflection  of  the  axis  is  made  in  the  proper  direction, 
by  means  of  the  levelling  screws  make  the  reading  of 
the  azimuth  level  (see  §  50)  the  same  after  deflection 
that  it  was  before  deflection,  and  there  will  then  be  no 
component  of  deflection  perpendicular  to  the  direction  P. 

45.  Example.  —  We  have  the  following  example  of 
the  record  and  reduction  of  the  first  half  of  a  complete 
set  of  observations  for  the  determination  of  d.  In  the 
reduction  we  note  that  the  divisor  2(6'  —  b")  of  Equation 
86  is  equivalent  to  26'  —  zb" ,  and  since  b  is  the  scale  read- 
ing of  the  middle  of  the  bubble,  26  is  equal  to  the  sum 
of  the  readings  of  the  ends  of  the  bubble.  The  column 
headed  26  is  found  in  this  way  from  the  mean  of  the  two 
sets  of  bubble  readings  opposite  each  circle  reading. 

The  regular  progression  of  the  numbers  in  the  column 
2(6'  —  6")  suggests  a  level-tube  of  variable  curvature, 
but  the  amount  of  data  is  not  sufficient  to  decide  this 
with  certainty.  More  observations  are  needed. 


110 


FIELD  ASTRONOMY. 


Friday,  Dec.  7,  1894. 

Alidade  Level  of  Universal  Instrument,  No.  2598. 
Readings  to  Mark. 

Mic.  I.  Mic.     II.  Mean. 

Axis  Vertical.  . 180°  26'  45"  26'    45  i8o°26'45' 

Axis  Deflected 179    27      o  27     6  179    27      3 


Azimuth  Circle. 
291°    o'o" 

IO  O 
20  O 

30  o 
40  o 
50  o 

2b      2(b'-b") 
26.45 
30.0 


38.5 

42.4 
46.75 


12  .05 
12  .40 

I2-75 

12  .40 


y  =       o    59  42 

Bubble. 
Forward.                          Backward. 

26.3         0.2                  26.0         0.4 
28.1          2.3                  27.7          1.9 
30.2         4.2                  29.7         3.9 
32.3        6.4              32.1        6.2 
34.0        8.1               34.2        8.5 
36.  2      10.  2              36  .  5      10.6 

A'  -A" 
log  tan  f 
log(A'-A") 

colog2(6'-6") 

30'  =  1800" 
8.   239 
3-  255 
8.   907 

log  d 
d 

o.  401 

2.    52 

46.  Inequality  of  Pivots. — When  a  spirit-level  is  used 
to  determine  the  inclination  of  a  line,  such  as  the  hori- 
zontal axis  of  a  transit,  its  readings  and  the  resulting 
inclination  will  be  vitiated  by  any  inequality  which  may 
exist  in  the  diameter  of  the  pivots  upon  which  it  rests. 
To  test  for  such  an  inequality  let  the  instrument  be 
firmly  mounted  and  the  inclination,  6',  be  measured 
with  the  level  as  shown  in  §  42 ;  then  lift  the  axis  out 
of  the  wyes,  turn  it  end  for  end,  and  replace  it  so  that 
what  was  the  east  pivot  shall  now  rest  in  the  west  wye. 
Again  measure  the  inclination,  b" ,  and  repeat  the  level- 
ings  and  reversals  several  times,  so  that  any  systematic 
difference  which  may  exist  between  bf  and  b"  shall  be 
well  determined.  We  now  put 

V>\  (87) 


INSTRUMENTS.  Ill 

where  i  is  the  correction  for  inequality  of  pivots,  and 
find  for  the  true  inclination  of  the  axis  in  the  two  posi- 
tions, 

b,  =  V-i,  b2  =  b"  +  i.  (88) 

The  correction  i  should  be  carefully  determined  and 
applied  to  all  measured  values  of  the  inclination. 

47.  The  Theodolite.  —  This  instrument,  which  is  also 
called  •  engineer's  transit,  altazimuth,  universal  instru- 
ment, etc.,  is  one  with  whose  general  appearance  and 
construction  the  student  is  supposed  sufficiently  familiar 
to  recognize  its  close  relationship  with  the  coordinates 
of  System  I,  altitude  and  azimuth.  Trace  out  this  rela- 
tionship in  Plates  I,  II,  and  III,  which  represent  different 
types  of  this  instrument.  The  line  of  sight  (telescope) 
is  a  radius  vector  of  undetermined  length;  the  hori- 
zontal and  vertical  circles  measure  azimuths  and  alti- 
tudes, or  zenith  distances,  and  in  an  ideally  perfect 
instrument  the  readings  of  these  circles  should  be  the 
true  azimuth  and  altitude  of  the  line  of  sight,  or  at 
most  should  differ  from  these  only  by  a  constant  index 
correction. 

It  may  readily  be  seen  that  among  the  conditions 
which  must  be  satisfied  in  the  construction  of  such  an 
instrument  are  the  following: 

(1)  The  axes  must  be  perpendicular  to  each  other. 

(2)  The  line  of  sight  must  be  perpendicular  to  the 
horizontal  axis. 

(3)  The  vertical  axis  must  be  truly  vertical. 
Owing  to  unavoidable  imperfections  of  mechanical 


112 


FIELD  ASTRONOMY. 


work  it  is  not  probable  that  any  one  of  these  conditions 
is  exactly  fulfilled  in  any  given  instrument,  and  they 
are  therefore  to  be  regarded  as  so  many  sources  of  error, 
whose  effects  may  be  made  small  by  careful  adjustment, 
but  whose  complete  elimination  must  be  sought  in  some 
other  way;  e.g.,  if  the  vertical  axis  is  not  truly  vertical, 
we  may  determine  as  follows  the  means  for  correcting 
the  effect  of  this  error  upon  the  measurement  of  altitudes. 
48.  Zenith  Distances. — In  Fig.  9  let  HZ  be  the  direc- 


FIG.  9. — Measurement  of  Zenith  Distances. 

tion  of  the  vertical;  5,  a  point  whose  zenith  distance, 
ZHS,  is  to  be  determined;  HV,  the  projection  of  the 
vertical  axis  of  the  instrument  upon  the  plane  HZS\ 


INSTRUMENTS.  113 

and  let  r'  denote  the  reading  of  the  vertical  circle  when 
the  line  of  sight  is  directed  toward  S.  After  reading  r' 
let  the  instrument  be  turned  about  the  vertical  axis 
through  an  angle  of  180°,  bringing  the  line  of  sight  into  the 
position  HS',  and  then  let  the  telescope  be  turned  about 
the  horizontal  axis  until  the  line  of  sight  again  points 
at  the  object  5,  and  let  r"  be  the  reading  of  the  vertical 
circle  in  this  position.  If  the  circle  is  numbered  in  quad- 
rants, as  is  very  common  in  small  instruments,  r'  and  r" 
will  be  approximately  the  same  number  but  with  a 
graduation  extending  from  o°  to  360°,  as  is  here  sup- 
posed, they  will  be  widely  different.  From  the  figure, 
the  angle  SHS'  is  measured  by  the  difference  of  these 
circle  readings,  r'  —  r"  ,  and  since  VHS  =  VHS',  we  have 
for  the  angular  distance  of  the  point  5  from  V  the 
equation, 

r").  (89) 


When  two  pointings  of  the  telescope  are  made  as 
above,  the  instrument  is  said  to  be  reversed  between  them, 
and  it  is  customary  to  designate  its  two  positions  as 
Circle  Right  and  Circle  Left,  respectively,  the  reference 
being  to  the  vertical  circle  of  the  instrument,  which 
faces  to  the  observer's  right  in  the  one  position  and  to 
his  left  in  the  other.  The  student  should  note  that  the 
angle  z"  is  here  determined  quite  independently  of  the 
adjustment  of  the  verniers,  which  may  be  intended  to 
read  altitudes,  zenith  distances,  or  anything  else,  since 
the  reversal  eliminates  all  question  of  adjustment  from 
the  difference  r'  —  r"  ,  and  is  made  for  this  purpose. 


114  FIELD  ASTRONOMY. 

The  true  zenith  distance  of  5  is,  however,  not  z"  but 
the  angle, 

z"+  b", 


and  b"  may  be  determined,  as  in  §  42,  from  readings 
of  the  spirit-level,  LL,  attached  to  the  instrument  in 
such  a  way  that  its  plane  is  parallel  to  the  line  of  sight, 
HS.  Such  a  level,  i.e.,  one  whose  tube  is  perpendicular 
to  the  horizontal  axis  of  the  instrument,  will  be  called 
the  altitude  level  of  the  instrument. 

A  convenient  method  of  taking  into  account  the 
readings  of  the  level-bubble  by  applying  them  directly 
to  the  circle  readings  instead  of  to  the  measured  angles, 
is  as  follows:  Let  w0  represent  the  reading  of  that  point 
of  the  level  scale  through  which  passes  that  radius  of 
the  level  which  is  parallel  to  the  vertical  axis,  HV,  CnQ 
in  the  figure,  and  let  n  denote  the  position  of  the  middle 
of  the  bubble  corresponding  to  the  circle  reading  /; 
i.e.,  since  the  bubble  always  stands  at  the  highest  point 
of  its  tube,  n  is  the  point  exactly  above  the  centre  of 
curvature,  C.  It  is  evident  from  the  figure  that 

b"  =  (n-nQ)2d  =  (a  +  b-  2  w0)d,  (90) 

where  d  is  the  value  of  half  a  level  division,  and  a  and  b 
are  the  actual  scale  readings  of  the  ends  of  the  bubble. 

If  the  instrument  had  been,  from  the  first,  perfectly 
levelled  we  should  not  have  obtained  rr  as  the  reading 
to  the  point  5,  but  in  place  of  r'  a  number  either  greater 
or  less  than  it  by  the  amount  b"  ;  and  if,  therefore,  we 
apply  to  r'  and  r"  level  corrections  determined  by  the 
equation  above  given  for  6",  we  shall  reduce  the  read- 


PLATE  III. 


A  German  Universal  Instrument.     Length  of  Horizontal  Axis 
12  inches.     Approximate  Cost  $400. 

\To  face  p.  114.] 


INSTRUMENTS.  115 

ings  to  what  they  would  have  been  for  a  perfectly  levelled 
instrument,  and  therefore  obtain  the  zenith  distance 
of  5  immediately  from  the  half  difference  of  the  corrected 
readings.  Since  any  constant  term  which  appears  in  the 
level  correction  will  be  eliminated  from  this  difference  of 
the  corrected  readings,  /  —  r",  we  may  substitute  in 
Equation  90,  in  place  of  2W0,  any  constant  number  what- 
ever, e.g.,  zero,  but  it  is  usually  convenient  to  take  as 
this  number  5,  the  total  number  of  divisions  included  in 
the  level  scale,  since  in  the  long  run  this  will  make  the 
level  corrections  small.  Making  this  substitution,  we 
have  finally, 

Level  Correction  =  ±  (a  +  b  —  S)d,  (91) 

where  the  ambiguous  sign  depends  upon  the  direction 
in  which  the  numbers  increase  along  the  level  scale, 
and  may  be  determined,  once  for  all,  for  a  given  instru- 
ment as  follows:  Two  readings  of  the  vertical  circle 
of  a  certain  instrument  were  taken  to  the  same  object, 
but  with  the  instrument  thrown  out  of  level  in  such  a 
way  that  the  bubble  stood  at  quite  different  parts  of 
the  scale  in  the  two  observations ;  e.g. : 

Observation.  Bubble.  Circle.  Level  Corr.       Corrected  r. 

a  b 

First 2.0     25.8     91°  9'     8'         +  i8"-7      91    9'   26". 7 

Second 7.9     31.9     91   9    40          —12.5       91   9     27. 5 

The  numerical  values  of  the  quantities  above  marked  Level  Corr. 
were  computed  from  Equation (91)  with  an  assumed  value  of  d  =  2ff.6, 
and  since  the  effect  of  these  corrections  must  be  to  bring  the  corrected 
circle  readings  into  agreement,  it  is  evident  that  the  +  sign  must  be 
used  for  the  first  observation  and  the  —  sign  for  the  second.  The 
whole  number  of  divisions  in  the  level  scale  being  35,  the  formula  for 
this  instrument  becomes, 

6"=+2".6  [35- (a  +  6)]. 


116  FIELD  ASTRONOMY. 

A  similar  formula  may  be  obtained  for  every  instrument,  and  a  table 
should  be  constructed  from  it,  which  with  the  argument  a+b  will 
show  the  value  of  b"  for  any  given  position  of  the  bubble.  Part  of 
such  a  table  is  given  below,  and  from  it  the  level  correction  correspond- 
ing to  any  ordinary  position  of  the  bubble  may  be  determined  by 
inspection. 

a  +  b  b"  a  +  b 

30  +i3"-o-          40 

31  +10    .4-          39 

32  +7-8-          38 

33  +5    .2-          37 

34  +    2  .6  —          36 

35  +o    .o-          35 

In  the  second  observation  given  above  we  have  a  +  b  =  7. 9  +  31.9=39.8, 
and  corresponding  to  this  number  we  find,  by  interpolation  from  the 
table,  b"  =  -12". 5. 

The  level  formulae  thus  derived  show  that  if  the 
bubble  be  brought  to  the  same  place  in  the  tube,  same 
values  of  a  and  b,  both  Circle  R.  and  Circle  L.,the  level 
correction  will  be  eliminated  from  the  difference  r'  —  r", 
and  may  therefore  be  neglected.  To  obtain  the  maxi- 
mum precision  however,  the  level  should  always  be 
read  and  a  correction  applied  to  each  circle  reading,  but 
even  when  this  is  done  it  is  good  practice  to  touch  up  the 
levelling  screws  after  each  reversal  and  bring  the  bubble 
back  as  near  as  may  be  to  its  first  position,  without,  how- 
ever, spending  too  much  time  in  obtaining  an  accurate 
agreement. 

In  some  instruments  the  level  and  verniers  are  at- 
tached to  a  frame  (alidade)  which  admits  of  rotation 
about  the  horizontal  axis  without  disturbing  the  direc- 
tion of  the  line  of  sight.  For  such  an  instrument  it 
may  be  shown  that  if,  before  reading  the  vernier,  the 
frame  be  turned  until  the  bubble  stands  at  the  middle 
of  *he  scale,  the  resulting  vernier  readings  are  equiva- 


INSTRUMENTS. 


117 


lent  to  the  corrected  circle  readings  derived  above,  and 
therefore  require  no  further  correction  for  level  error. 
This  mechanical  device,  although  convenient  for  some 
purposes,  is  of  inferior  accuracy. 

49.  Effect  of  Errors  of  Adjustment.  —  A  geometrical 
investigation  similar  to  the  above  may  be  made  to  show 
the  effect  of  each  source  of  instrumental  error,  but  we 
shall  find  it  more  convenient  to  develop  the  combined 
effect  of  these  errors  through  an  analysis  based  upon 
Fig.  10,  which  represents  a  part  of  the  celestial  sphere, 


FIG.  io.— Theory  of  the  Theodolite. 

where  Z  is  the  zenith,  V  is  the  point  in  which  the  vertical 
axis  of  the  instrument,  when  produced,  cuts  the  sphere,, 
H  is  the  point  of  the  sphere  determined  by  the  prolonga- 
tion of  the  horizontal  axis,  and  5  is  a  star  or  other  object 
whose  azimuth  and  altitude  are  to  be  determined  from 
readings  of  the  horizontal  and  vertical  circles  of  the 
instrument.  The  angles  measured  by  means  of  these 


118  FIELD  ASTRONOMY. 

circles  lie  in  the  planes  of  the  circles,  but  just  as  the 
azimuth  of  a  point  is  measured  either  by  an  arc  of  the 
horizon  or  by  the  corresponding  spherical  angle  at  the 
zenith,  so  the  data  furnished  by  the  vernier  readings 
may  be  regarded  as  spherical  angles  having  their  ver- 
tices respectively  at  V  and  H.  Thus  if  r  represent  the 
reading  of  the  vertical  circle  when  the  line  of  sight  is 
directed  toward  5,  and  r0  is  the  reading  when  this  line 
is  directed  toward  some  point  in  the  arc  HV,  the  differ- 
ence, r  —  r0,  measures  the  spherical  angle  VHS.  Simi- 
larly, for  the  horizontal  circle,  by  rotating  the  instrument 
about  its  vertical  axis,  H  may  be  moved  from  its  present 
position,  corresponding  to  the  reading  R,  into  a  new 
position  falling  upon  the  arc  VM,  and  if  Rt  be  the  circle 
reading  in  this  position,  we  shall  find  that  R  —  Rl  equals 
the  spherical  angle  HVM.  From  these  spherical  angles, 
determined  by  the  circle  readings,  it  is  required  to  find 
the  true  direction,  MZS  =  Af,  and  the  true  zenith  dis- 
tance, ZS  =  2,  of  the  star  5. 

It  is  evident  from  the  figure  that  the  arc  VH  =  90°  —  i, 
measures  the  angle  between  the  vertical  and  the  horizontal 
axis  of  the  instrument,  and  that  i  is  therefore  the  error 
of  adjustment  of  the  axes,  corresponding  to  Condition  i, 
§  47.  Similarly,  HS  =  90°  +  c  measures  the  angle  be- 
tween the  horizontal  axis  and  the  line  of  sight,  and  c  is 
the  error  in  the  adjustment  corresponding  to  Condition  2. 
Also,  VZ  =  ?  is  the  error  of  level  of  the  instrument, 
i.e.,  deviation  of  the  vertical  axis  from  the  true  vertical, 
corresponding  to  Condition  3.  The  arc  HZ  =  go°  —  b 
measures  the  angle  that  the  horizontal  axis  makes  with 


INSTRUMENTS.  119 

the  true  vertical,  and  b  is  therefore  the  level  error  of  this 
axis.  Note  that  as  the  instrument  is  turned  into  differ- 
ent positions  by  rotation  about  the  axes  V  and  H,  the 
quantities  7-,  i,  and  c  remain  unchanged  and  are  there- 
fore called  instrumental  constants,  since  they  define 
the  condition  of  the  instrument  with  respect  to  its 
several  adjustments.  The  level  error,  b,  is  sometimes 
included  among  these  constants,  but  is  not  strictly  one 
of  them,  since  its  value  changes  as  the  instrument  is 
turned  in  azimuth. 

We  shall  suppose  the  instrument  to  be  so  well  ad- 
justed that  none  of  the  instrumental  constants  exceeds 
2',  and  H  will  then  be  so  near  the  pole  of  the  great  cir- 
cle VZM  that  we  may  assume  without  sensible  error 
HVM=HZM  and,  replacing  these  quantities  by  their 
equivalents,  obtain 


or 

A^R-(Rl  +  go°)-w.  (92) 

The  azimuth  of  5,  reckoned  from  the  true  meridian 
instead  of  from  the  arc  VM,  differs  from  A'  only  by 
the  substitution  of  another  constant,  the  index  correc- 
tion of  the  horizontal  circle,  in  place  of  7?t  +  90°  ;  and  as 
this  index  correction  must  in  any  case  be  separately 
determined  (see  §38),  we  may  replace  the  constant  term 
7^  +  90°  by  R0,  the  index  correction  referred  to  the  true 
meridian,  and  we  shall  then  have  for  the  true  azimuth  of  5, 

A-R-R.-W.  (93) 


120  FIELD  ASTRONOMY. 

The  auxiliary  quantity  w  has  thus  far  been  defined 
only  by  means  of  Fig.  10,  where  the  spherical  angle  HZS 
is  labelled  90°+  w.  To  determine  the  value  of  w  in  terms 
of  the  instrumental  constants,  we  have  from  the  triangle 
HZS  by  means  of  Equations  4,  the  relation, 

—  sin  c  =  sin  b  cos  z  —  cos  b  sin  z  sin  w, 

which,  since  b  and  c  are  small  quantities,  is  equivalent  to 

c  b 


sin  z     tan  z9 

or,  replacing  the  zenith  distance,  z,  by  the  star's  alti- 
tude, h, 

w  =  c  sec  h  +  b  tan  h.  (94*) 

Since  neither  i  nor  7-  enters  into  this  equation,  the  effect 
of  these  errors  must  be  taken  into  account  through  6, 
the  inclination  of  the  horizontal  axis.  This  is  to  be  deter- 
mined with  a  spirit-level,  and  each  circle  reading,  Rt 
must  be  corrected  for  the  particular  inclination  of  the 
axis  that  corresponds  to  R.  The  factor  tan  h  becomes 
zero  for  an  object  in  the  horizon,  and  for  this  special  case 
the  effect  upon  the  azimuth  readings  of  an  error  of  level 
is  zero.  On  the  other  hand,  when  the  object  to  be 
observed  is  at  a  considerable  elevation,  e.g.,  the  Pole 
Star  in  an  azimuth  determination,  the  factor,  tan  h, 
becomes  large  and  the  effect  of  level  error  is  magnified. 
It  is  in  fact  one  of  the  chief  sources  of  error  in  such  deter- 
minations. 

50.  Determination  of  Errors  of  Adjustment. — The  error 
above  represented  by  c  is  called  the   collimation,  and 


INSTRUMENTS.  121 

its  effect  is  usually  to  be  eliminated  through  a  reversal 
of  the  instrument.  Since  the  angular  distance  of  5  from 
one  end  of  the  horizontal  axis  is  90°  +  c,  its  distance 
from  the  other  end  must  be  90°  —  c,  and  as  in  the  rever- 
sal these  ends  change  places  the  effect  of  c  must  have 
one  sign  Circle  R.,  and  the  opposite  sign  Circle  L.,  and 
will  therefore  be  eliminated  from  the  mean  of  observa- 
tions taken  in  both  positions. 

In  precisely  the  same  way  it  may  be  shown  that  the 
effect  of  i,  error  of  adjustment  of  the  axes,  is  eliminated 
from  the  mean  of  observations  taken  in  the  two  posi- 
tions, and  wherever  any  considerable  precision  is  required 
in  azimuth  observations  or  in  the  measurement  of  hori- 
zontal angles,  the  observer  should  not  fail  to  make  an 
equal  number  of  pointings  in  each  position  of  the  instru- 
ment to  secure  this  elimination  of  errors. 

In  the  triangle  HVZ  the  angle  HVZ=HZM  is  very 
nearly  equal  to  90° +  ^4',  and  assuming  this  equality  we 
find  from  this  triangle, 

sin  b  =  sin  i  cos  ?  —  cos  i  sin  7-  sin  A',  (95) 

which  is  equivalent  to, 

b=i-rsinA'.  (96) 

The  quantity  7-  sin  A',  which  we  shall  represent  here- 
after by  the  symbol  &',  and  which  corresponds  to  the 
arc  ZI  of  Fig.  10,  is  that  component  of  the  level  error 
of  the  vertical  axis,  7-,  which  lies  at  right  angles  to  the 
line  of  sight  and  which  may  therefore  be  determined 
from  the  readings  of  a  level  parallel  to  the  horizontal 


122  FIELD  ASTRONOMY. 

axis  of  the  instrument.  Such  a  level  is  called  the  azi- 
muth level,  and  if  resting  upon  the  axis  and  capable  of 
reversal  (striding-level) ,  it  is  most  conveniently  used 
to  determine  the  level  error  of  this  axis,  b.  If  fastened 
to  the  frame  of  the  instrument  and  incapable  of  reversal, 
it  may  be  used  to  determine,  from  bubble  readings  taken 
Circle  R.  and  Circle  L.,  the  value  of  bf  for  the  vertical 
axis,  and  corresponding  to  these  two  cases  we  shall  have 
the  following  expressions  of  the  level  corrections  to  be 
applied  to  readings  of  the  horizontal  circle : 

Striding-level,    —  6  tan  h.     Both  Circle  R.  and  Circle  L. 
Fixed  Level,       —  b'  tan  h.    Mean  of  Circle  R.  and  L. 

If,  as  is  usual,  the  graduation  of  the  circle  increases  from 
left  to  right,  b  and  b'  are  to  be  considered  essentially 
positive  when  the  high  end  of  the  horizontal  axis  has  an 
azimuth  90°  greater  than  the  object  5. 

The  student  should  not  fail  to  note  in  connection  with 
the  use  of  a  fixed  azimuth  level  that  if  the  bubble  is 
brought  to  the  same  scale  reading,  Circle  R.  and  Circle  L., 
b'  will  be  zero  and  the  level  error  will  be  eliminated  from 
the  mean  result. 

A  reversal  furnishes  a  convenient  method  for  deter- 
mining or  adjusting  the  collimation.  For  this  purpose 
let  Rf  and  R"  be  readings  of  the  horizontal  circle  corre- 
sponding to  observations  of  a  fixed  mark  in  or  very  near 
the  horizon,  made  in  the  two  positions  of  the  instrument ; 
then,  from  Equations  92  and  94, 

2C  =  R'-R".  (97) 


INSTRUMENTS.  123 

To  determine  the  error  of  adjustment  of  the  axes,  i,  let 
the  inclinations  of  the  horizontal  axis,  bv  b2,  be  meas- 
ured in  two  positions  of  the  instrument  differing  180° 
in  azimuth,  i.e.,  when  Vernier  A  reads  o°  and  when  it 
reads  180°.  We  shall  then  have,  from  Equation  96, 

bi=i—  j-  sin  A', 

b2=i-  r  sin  (A'  +  180°)  =i+  r  sin  A', 

from  which  we  obtain  immediately 

2i  =  bi  +  b2.  (98) 

If  we  call  the  inclinations  bv  b2  positive  when  the  circle 
end  of  the  axis  is  too  high,  a  positive  value  of  i  will  indi- 
cate that  the  same  end  is  too  high,  i.e.,  it  makes  too 
small  an  angle  with  the  upward  extension  of  the  vertical 
axis. 

The  value  of  7-,  which  will  seldom  be  required,  may 
be  found  from  four  values  of  b  determined  at  intervals 
of  90°  in  azimuth. 

51.  Additional  Theorems.  —  By  an  analysis  similar  to 
that  employed  above,  it  may  be  shown  from  the  trian- 
gles HSZ,  HZV,  of  Fig.  10,  that  the  errors  b,  c,  and  i  have 
no  appreciable  influence  upon  observations  of  altitude 
or  zenith  distance.  Indeed,  it  may  be  seen  without 
formal  analysis  that  when  c,  b,  and  i  are  small  quantities, 
H  is  so  nearly  the  pole  of  the  circles  ZS,  VS,  that  these 
arcs  are  measured  by  the  corresponding  angles  at  H, 
i.e.,  by  the  readings  of  the  vertical  circle  unconnected 
for  instrumental  error.  Since  the  error  corresponding 
to  7-  is  taken  into  account  in  the  approximate  analysis 


FIELD  ASTRONOMY. 

of  §  48,  we  may  adopt  as  definitive  the  results  there 
obtained.  The  correction  b"  there  determined  is  the 
arc  VI  of  Fig.  10,  i.e.,  it  is  the  projection  of  7-  upon  the 
line  of  sight,  VS. 

The  demonstration  of  the  following  theorems,  which 
are  of  some  consequence  in  the  use  of  a  theodolite,  is 
left  to  the  student. 

i.  If,  as  is  quite  common  in  engineer's  transits,  the 
vertical  circle  is  graduated  into  quadrants  instead  of 
from  o°  to  360°,  observations  of  altitude  should  be  made 
in  the  way  already  indicated,  but  in  their  reduction 
we  shall  have,  in  place  of  the  formula  for  2",  the  substi- 
tute, 

,  (99) 


i.e.,  the  mean  of  the  readings  gives  directly  the  instru- 
mental altitude. 

2.  The  altitude  level  of  such  an  instrument  usually 
has  the  zero  of  its  scale  placed  at  the  middle  of  the  tube, 
and  when  such  is   the  case  readings  of  that  end  of  the 
bubble  nearest  the  objective  end  of  the  telescope  should 
be  marked  o,  and  those  of  the  end  nearest  the  eyepiece 
should  be  called  e\  the  formula  for  level  correction  then 
becomes, 

b"  =  (o-e)d.  (100) 

3.  A  theodolite  maybe  reversed  by  lifting  the  tele- 
scope from  its  supports,  turning  the  axis  end  for  end, 
and  replacing  it  in  the  wyes  in  the  changed    position. 
This  mode  of  reversal  eliminates  errors  of  level  and  colli- 
mation  quite  as  well  as  does  the  one  above   described, 


INSTRUMENTS.  125 

and  also  eliminates  the  inequality  of  pivots  from  the 
determination  of  b.  It  is  therefore  to  be  preferrecl 
when  it  can  be  conveniently  practised. 

52.  Errors  Arising  from  the  Circle  Readings.  —  Numerous 
errors  of  a  class  not  considered  above,  creep  into  the 
results  of  observation  through  the  circle  readings,  which 
may  be  vitiated  in  greater  or  less  degree  by: 

(a)  Defective  graduation  of  the  circle  itself. 

(b)  The  plane  of  the  circle  not  being  normal  to  the 

rotation  axis. 

(c)  The  circle  not  being  truly  centred  upon  the  axis. 

(d)  The  spaces  on  the  vernier  being  too  large  or  too 

small  relative  to  those  on  the  circle. 

(e)  Error  of  focussing  (runs)  in  the  reading  micro- 

scopes, 
etc.  etc.  etc. 

The  detailed  study  of  these  sources  of  error  lies  be- 
yond the  scope  of  the  present  work,  but  we  note  that  in 
great  part  their  effects  may  be  eliminated  by  taking  the 
mean  of  a  considerable  number  of  observations  in  which 
the  circle  readings  are  symmetrically  distributed  through- 
out the  whole  360°  of  the  graduation.  Thus  if  an  angle 
of  120°  between  objects  A  and  B  is  measured  three  times 
and  the  circle  turned  120°  after  each  measurement  so 
as  to  obtain  the  following  system  of  readings : 

To  A.  To  B.  P-A. 

Observation  i o°  o'  o"  120°  o'  o"   120°  o'  o" 

2 I2O   O   O    240   O   O    I2O   O   O 

"      3 240  o  o  360  o  o   120  o  o 


126  FIELD  ASTRONOMY. 

whatever  graduation  errors  may  affect  the  particular 
reading  120°  o'  oo"  will  be  eliminated  from  the  mean 
value  of  B  —  A,  since  this  reading  enters  into  that  mean 
once  with  a  plus  sign  and  once  with  a  minus  sign.  If 
the  required  angle  is  small,  e.g.,  i°,  it  will  not  be  con- 
venient to  carry  out  the  above  programme  of  reading 
around  the  entire  circle,  but  the  elimination  of  errors 
may  still  be  made  by  shifting  the  circle  so  that  the 
readings  to  object  A  may  be  symmetrically  distributed 
through  the  entire  circumference,  e.g.,  every  60°  or 
every  30°.  For  an  instrument  provided  with  two  ver- 
niers or  microscopes  it  will  suffice  to  distribute  the  read- 
ings of  each  vernier  over  an  arc  of  180°. 

53.  The  Method  of  Repetitions.  —  A  peculiar  method 
of  measuring  horizontal  angles  may  be  adopted  with 
advantage  if,  as  is  often  the  case,  the  instrument  is 
provided  with  two  motions  in  azimuth  called,  respectively, 
upper  and  lower,  one  of  which  produces  a  change  in  the 
vernier  readings,  while  in  the  other,  verniers  and  circle 
remain  firmly  clamped  together  and  turn  simultaneously, 
without  change  in  the  circle  reading.  Reverting  to 
§  52,  we  may  note  that  the  circle  readings  120°,  240°, 
there  recorded,  are  quite  unnecessary  since,  if  the  first 
reading,  o°,  be  subtracted  from  the  last  one,  360°,  and 
the  result  divided  by  3,  we  shall  have  as  the  value  of 
the  angle  i2o°o'o",  which  is  precisely  the  same  as 
the  mean  of  the  three  values  of  B  —  A,  and  is  all  that 
that  mean  can  furnish. 

This  process  is  called  the  method  of  repetitions  and 
consists,  essentially,  in  making  a  series  of  pointings  upon 


INSTRUMENTS.  127 

two  objects  between  which  an  angle  is  to  be  measured, 
turning  always  from  A  to  B  upon  the  upper  motion  of 
the  instrument  and  from  B  to  A  upon  the  lower  motion, 
so  that  the  vernier  reading  in  the  latter  turning  is  not 
changed.  A  series  of  such  pointings  is  called  a  set  and 
the  verniers  need  be  read  only  for  the  first  and  last 
pointings  of  the  set.  If  the  initial  and  final  readings  be 
represented  by  R'  and  R" ',  and  n  be  the  number  of  point- 
ings to  each  object  contained  in  the  set,  we  shall  have,  as 
shown  above, 

T)f  D// 

Angle  =  -— — .  (101) 

It  is  often  advantageous  to  reverse  the  instrument  at 
the  middle  of  a  set,  turning  on  the  lower  motion,  and  thus 
secure  an  additional  elimination  of  instrumental  errors. 

The  advantages  of  the  method  of  repetitions  are  a 
saving  of  labor  through  the  diminished  number  of  vernier 
readings  and,  where  the  verniers  are  comparatively 
coarse,  an  increase  of  accuracy  through  the  introduction 
of  the  divisor  n  into  the  value  of  the  angle.  The  pre- 
cision of  a  small  instrument,  such  as  an  engineer's  transit, 
may  be  considerably  increased  in  this  way,  but  for  the 
larger  instruments,  provided  with  micrometer  micro- 
scopes, experience  shows  that  the  best  results  are  to  be 
obtained  by  reading  the  microscopes  after  every  pointing. 

Where  a  horizontal  angle  between  objects  at  very- 
different  altitudes  is  to  be  measured  by  the  method  of 
repetitions,  as  in  an  azimuth  determination,  an  addi- 
tional source  of  error  requires  careful  attention,  viz.,  the 
effect  of  a  lack  of  parallelism  between  the  axes  corre- 


128  FIELD  ASTRONOMY. 

spending  to  the  upper  and  lower  motions  of  the  instru- 
ment. To  eliminate  this  error  we  proceed  in  the  follow- 
ing manner:  The  axis  of  the  lower  motion  should  be 
made  as  nearly  vertical  as  possible,  and  whatever  may 
be  the  error  of  the  upper  axis  it  will  produce  no  effect 
upon  the  final  result  if  the  number  of  repetitions  is  so 
chosen  that  the  set  extends  through  360°;  for  in  the 
successive  turnings  about  the  lower  motion  the  upper 
axis  has  been  made  to  describe  a  complete  cone  about 
the  lower  axis,  and  any  error  which  may  have  been  caused 
by  a  deflection  to  the  east  in  one  part  of  the  set  is  bal- 
anced by  the  opposite  error,  caused  by  a  deflection  to 
the  west,  in  another  part,  etc.  If  the  angle  to  be  meas- 
ured is  so  small  that  the  set  cannot  be  made  to  extend 
through  360°,  the  following  observing  programme  will 
also  eliminate  the  error  of  the  axis :  Measure  a  set  of  any 
desired  number  of  repetitions.  When  it  is  completed 
leave  the  instrument  clamped  at  the  last  vernier  read- 
ing, reverse  about  the  lower  motion  and  repeat  the  set  in 
the  opposite  direction,  i.e.,  beginning  with  the  object 
last  sighted  upon  and  with  approximately  the  vernier 
reading  last  obtained. 

The  level  correction  to  the  circle  readings  should  be 
derived  in  the  ordinary  way,  §  50,  from  readings  of  a 
level  taken  when  the  instrument  is  reversed  about  the 
lower  axis. 

54.  Precepts  for  the  Use  of  a  Theodolite.  —  The  ex- 
perience of  the  principal  geodetic  surveys  indicates  that 
the  following  precepts  should  be  observed  in  all  precise 
work  with  a  theodolite : 


INSTRUMENTS.  129 

(1)  An   equal  number  of  measurements   should  be 
made  in  each  position  of  the  instrument,  Circle  R.  and 
Circle  L. 

(2)  An  equal  number  should  be  taken  in  each  direc- 
tion, i.e.,  the  line  of  sight  turned  from  right  to  left  and 
from  left  to  right. 

(3)  The  position  of  the  circle  should  be  so  shifted  from 
time  to  time  that  the  readings  to  each  object  are  sym- 
metrically distributed  throughout  the  360°. 

(4)  The  observations  should  be  made  as  rapidly  as 
the  observer  can  work  without  undue  haste. 

55.  The  Sextant.  —  A  sextant  consists  essentially  of 
two  mirrors  and  a  graduated  arc  of  a  circle,  about  60°, 
for  measuring  the  angle  between  the  planes  of  the  mir- 
rors. The  peculiar  value  of  the  instrument  lies  in  the 
fact  that  it  is  light  and  portable,  requires  no  fixed  support, 
and  may  therefore  be  used  for  the  measurement  of 
angles  at  sea  as  well  as  on  shore,  and  in  any  plane,  ver- 
tical, horizontal,  or  inclined.  For  the  purpose  of  de- 
scription and  analysis  we  suppose  the  sextant  to  be 
placed  upon  a  table,  with  the  plane  of  its  arc  horizontal, 
and  we  shall  use  the  terms  altitude,  azimuth,  etc.,  with 
reference  to  this  special  position  of  the  instrument.  The 
conclusions  drawn  from  this  consideration  of  the  in- 
strument apply  equally  when  it  is  used  in  any  other  plane. 

The  essential  parts  of  a  sextant  are  indicated  in 
Fig.  ii  which  should  be  compared  with  Plate  IV.  At 
the  centre  of  the  arc  is  a  vertical  axis  carrying  a 
vernier-arm,  V,  and  also  supporting  one  of  the  mirrors 
called  the  index-glass,  I,  whose  plane  is  vertical,  passes 


130  FIELD  ASTRONOMY. 

nearly  through  the  axis  and  rotates  with  the  vernier- 
arm  as  the  latter  is  turned  in  azimuth.  At  one  side  of 
the  sextant  frame  is  the  other  mirror,  H,  called  the  hori- 
zon-glass, with  its  plane  vertical  and  fixed  parallel  to 
that  radius  of  the  graduated  arc  which  is  numbered  o°. 
Only  the  lower  half  of  the  horizon-glass  is  silvered,  the 
upper  half  is  left  transparent.  A  telescope,  T,  is  mounted 
on  the  side  of  the  frame  opposite  to  the  horizon-glass 
and  has  its  line  of  sight  directed  toward  the  latter. 
From  Fig.  n  it  may  be  seen  that  an  observer  looking 


FIG.  ii. — Elements  of  a  Sextant. 

into  the  telescope  and  through  the  unsilvered  upper  half 
of  the  horizon -glass  will  see  that  part  of  the  horizon 
toward  which  the  telescope  is  directed,  and  will  also  see 
superposed  upon  it  a  view  of  another  part  of  the  horizon 
reflected  from  the  index -glass  to  the  silvered  half  of  the 
horizon -glass,  and  from  this  again  reflected  into  the  tele- 
scope. This  part  of  the  horizon  is  said  to  be  seen 


| 


INSTRUMENTS.  131 

reflected,  while  the  part  seen  through  the  horizon-glass  is 
observed  direct.  Any  reflected  image  which  is  super- 
posed upon  a  direct  image  is  said  to  be  in  contact  with 
the  latter,  and  we  shall  represent  these  images  as  seen 
in  the  telescope,  by  /  and  H  respectively. 

By  turning  the  index -glass  in  azimuth,  different  parts 
of.  the  horizon  may  be  reflected  into  the  telescope,  and 
since  the  rays  of  light  incident  upon  and  reflected  from 
the  mirror  make  equal  angles  with  its  surface,  it  is 
apparent  that  for  every  i°  that  the  mirror  is  turned,  the 
azimuth  of  the  point  reflected  into  the  telescope  will 
be  changed  by  2°.  There  may  be  found  by  trial  a  set- 
ting of  the  index -glass  at  which  both  a  direct  and  a  re- 
flected image  of  the  same  object  may  be  seen  simul- 
taneously and  may  be  made  to  pass  one  over  the  other 
as  the  vernier-arm  is  slightly  turned.  Let  R0  denote 
the  vernier  reading  when  these  images  are  brought  into 
contact,  and  let  R  be  the  reading  at  which  any  other 
object,  /,  is  brought  into  contact  with  the  H  just  ob- 
served; then  it  appears  from  the  above  that  the  differ- 
ence of  azimuth  between  /  and  H  is  twice  the  angle 
included  between  RQ  and  R.  On  account  of  this  mul- 
tiplier, 2,  each  half -degree  of  the  sextant  arc  is  numbered 
as  if  it  were  a  whole  degree,  and  we  have,  therefore,  for 
the  difference  of  azimuth, 

PI-I  =  R-RQ.  (I02) 

The  —  R0  which  appears  in  this  equation  is  called  the 
index  correction,  and  it  should  be  observed  that,  owing 
to  the  angle  subtended  at  the  object  H  by  the  space 


132  FIELD  ASTRONOMY. 

separating  the  index  and  horizon  glasses,  the  reading  RQ 
will  depend  upon  the  distance  of  H  from  the  instrument. 
If  it  is  near  at  hand,  less  than  three  miles,  RQ  should  be 
determined  as  above  and  the  axis  of  rotation  of  the 
index -glass  should  be  centred  over  the  point  at  which  it 
is  desired  the  vertex  of  the  measured  angle  should  fall. 
If  the  objects  are  very  remote,  all  question  of  the  exact 
position  of  the  vertex  is  eliminated,  and  a  mode  of  deter- 
mining the  index  correction  given  hereafter  will  be  found 
more  convenient  than  the  above. 

It  may  now  be  seen  that  the  following  conditions 
must  be  satisfied  in  order  that  the  Equation  102,  given 
above,  shall  furnish  the  true  value  of  the  angle  between 
H  and  /: 

(a)  The  rotation  axis  and  the  plane  of  the  index -glass 
must  be  perpendicular  to  the  plane  of  the  graduated 
arc.     If   they    are   not    perpendicular,  this    arc    cannot 
accurately  measure  the  amount  of  rotation  of  the  mirror. 

(b)  The  horizon-glass  must  be  perpendicular  to  the 
plane  of  the  arc.     If  it  is  not  perpendicular,  the  direct 
and  reflected  images  of  H  cannot  be  brought  into  con- 
tact,.but  one  will  pass  above  or  below  the  other  as  the 
vernier-arm  is  turned. 

(c)  The  objects  H  and  /  must  lie  in  the  sextant  hori- 
zon, for  otherwise  the  difference  of  their  azimuths  would 
not   be    the    true    angle    between    them.     The    sextant 
horizon  must  here  be  understood  to  mean  the  plane  of 
the  graduated  arc,  and  this  condition  will  be  satisfied 
if  the  sextant  is  so  held  during  the  observation  that  this 
plane  passes  through  the  objects  H  and  7. 


INSTRUMENTS.  133 

56.  Adjustments  of  the  Sextant. — (A)  The  Index -glass. 
— Take  the  telescope  out  from  its  support,  set  it  on 
end  at  any  part  of  the  arc,  and  turn  the  index -glass 
until  its  plane  passes  a  little  to  one  side  of  the  telescope. 
By  holding  the  eye  a  little  to  the  right  of  the  line  joining 
the  index -glass  to  the  telescope  a  reflected  image  of  the 
telescope  may  be  seen  simultaneously  with  a  direct  view 
of  it,  and  these  two  images  should  be  parallel,  provided 
the  telescope  stands  normal  to  the  plane  of  the  arc. 
Any  error  in  this  last  condition  may  be  eliminated  by 
turning  the  telescope  180°  about  its  own  axis  and  repeat- 
ing the  test.  No  ad  jus  ting -screws  are  provided  for  the 
index-glass,  but  it  may  be  adjusted,  if  necessary,  by  re- 
moving it  from  its  frame  and  filing  down  the  bearing- 
points  against  which  it  is  held. 

(B)  The  Horizon- glass. — Bring    the    direct    and    re- 
flected images  of  a  distant  object  into  contact  if  possible. 
If  this  cannot  be  done,  bring  them  near  together  and 
tilt  the  horizon -glass  by  means    of  its    adjusting  screws 
until  by  turning  the  vernier -arm  the  images  can  be  made 
to  coincide. 

(C)  The  Telescope. — To  enable  the  observer  to  make 
the  plane  of  the  sextant  pass  through  the  objects  H 
and  /  it  is  customary  to   place  in  the  eyepiece   of  the 
telescope  a  pair  of  coarse  threads  which  should  be  set 
parallel  to  the  plane  of  the  sextant.     By  means  of  its 
adjusting  screws  the  telescope    should   be  tilted  up  or 
down  until  the  line  of  sight  passing  midway  between 
these  threads  is  parallel  to  the  plane  of  the  sextant.    If 
the  objects  H  and  /  are  brought  midway  between  these 


134  FIELD  ASTRONOMY. 

threads  when  contact  between  them  is  made,  they  will 
lie  in  the  plane  of  the  sextant  as  required.  To  deter- 
mine if  the  telescope  is  properly  tilted,  select  two  well- 
defined  objects  about  120°  apart,  and  bring  them  into 
contact  when  the  sextant  is  so  held  that  they  are  both 
seen  in  the  upper  part  of  the  field  of  view.  Then  shift 
the  position  of  the  sextant  plane  so  as  to  bring  the 
objects  to  the  lower  part  of  the  field  and  note  whether 
they  remain  in  contact  or  appear  separated;  if  they  are 
appreciably  separated  the  telescope  requires  further 
adjustment. 

57.  Outstanding  Errors  of  the  Sextant. — The  methods 
of  adjustment  above  described  are  only  approximate,  and 
the  readings  of  the  instrument  will  be  affected  by  what- 
ever error  remains  in  the  adjustment.  In  general  the 
effect  of  these  errors  will  be  small  for  small  angles,  but 
will  increase  rapidly  with  the  magnitude  of  the  angle 
measured,  and  the  adjustments  should  be  made  correct 
to  within  10'  if  the  resulting  errors  for  an  angle  of  90° 
are  to  be  insensible. 

However  carefully  these  adjustments  are  made  there 
will  remain  a  source  of  error  which  cannot  be  removed 
by  adjustment,  but  whose  effect  must  be  determined 
and  applied  as  a  correction  to  the  readings  if  the  maxi- 
mum attainable  precision  of  the  instrument  is  required. 
It  is  assumed  above  that  the  centre  of  the  graduated  arc 
falls  exactly  at  the  centre  of  motion  of  the  index -glass, 
but  the  maker  is  seldom  able  to  secure  this  exact  agree- 
ment, and  without  it  the  readings  of  the  vernier  are  not 
an  accurate  measure  of  the  amount  of  rotation  of  the 


INSTRUMENTS.  135 

mirror.  The  effect  of  this  error,  which  is  called  eccen- 
tricity, combined  with  the  effect  of  all  other  outstanding 
errors  of  the  instrument,  is  best  determined  by  carefully 
measuring  with  it  a  set  of  known  angles  of  different 
magnitudes,  from  o°  to  the  largest  one  possible,  and  treat- 
ing the  difference  between  the  measured  value  and  the 
true  value  of  each  angle  as  a  correction  to  the  corre- 
sponding reading  of  the  sextant.  These  corrections  may 
be  plotted  as  ordinates  with  the  sextant  readings  as 
abscissas  and  a  curve  drawn,  from  which  intermediate 
values  of  the  correction  may  be  read.  The  length  of 
the  arc  joining  two  stars  whose  right  ascensions  and 
declinations  are  given,  may  be  computed  and  used  as  a 
known  angle  for  this  purpose,  provided  the  effect  of 
refraction  in  altering  this  distance  is  duly  taken  into 
account;  or  if  a  distant  part  of  the  horizon  can  be  seen, 
a  set  of  angles  may  be  measured  with  a  good  theodolite 
for  comparison  with  the  sextant  results. 

58.  Index  Correction. — Since  the  value  of  the  index 
correction  for  very  distant  objects  is  constant  so  long 
as  the  adjustments  of  the  sextant  remain  unchanged, 
it  may  be  determined  from  special  observations  made 
for  this  purpose,  but  the  determination  should  be  fre- 
quently repeated  since  the  adjustment  is  easily  disturbed. 
Let  a  shade -glass  be  placed  over  the  eye  end  of  the  tele- 
scope and  the  direct  and  reflected  images  of  the  sun 
brought  into  contact,  externally  tangent  to  each  other, 
in  each  of  the  two  possible  positions,  H  first  right,  then 
left  of  /.  The  mean  of  the  corresponding  sextant  read- 
ings will  be  the  required  value  of  RQ.  Since  the  index 


136  FIELD  ASTRONOMY. 

correction  enters  into  the  value  of  every  measured  angle, 
it  should  be  carefully  determined  from  several  settings, 
as  in  the  following  example: 

DOUBLE   DIAMETER    OF   SUN   FOR   INDEX    CORRECTION. 

Caroline  Island,  April  22,  1883. 

Observer,  W.  U. 

On  Arc.  Off  Arc.  Reduction. 

o°  26'     o"  359°  22'     5"  #0  =  359°  53'   54" 

26       o  220  i=          +66 

26       5  22       o 

26       o  21     40  45=      i       4       o 

25    40  21    50  5=     0160 

25    40  21     50  Almanac  =      o     15    56.4 


360°  25'   54" 

In  place  of  subtracting  RQ  from  each  subsequent 
reading  of  the  instrument  it  is  in  this  case  more  con- 
venient to  employ  the  quantity  ^  =  360°  —  R0  as  a  correc- 
tion to  be  added.  The  readings  ' '  Off  Arc  "  were  taken  on 
the  supplementary  arc  to  the  right  of  the  o°,  and  the 
student  should  note  that  the  resulting  R0  falls  off  the  arc. 
Referring  to  the  position  of  RQ,  the  sign  of  the  index 
correction  may  be  determined  from  the  sailor's  rule: 
When  it  (RQ)  is  on  it's  off  (i  subtr active] ,  and  when  it  (R0) 
is  off  it's  on  (i  additive}.  The  values  of  the  sun's  semi- 
diameter  furnished  by  the  observations  and  given  in  the 
almanac  are  shown  above  for  comparison. 

59.  Artificial  Horizon.  —  Altitudes  may  be  measured 
with  a  sextant  either  from  the  natural  (sea)  horizon  or 
from  an  artificial  horizon,  one  form  of  which  is  a  shallow 
box  containing  mercury,  covered  by  a  glass  roof  to  pro- 
tect it  from  wind.  The  reflecting  surface  of  the  liquid 
is  improved  by  adding  to  it  a  little  tin-foil  and  removing 


INSTRUMENTS.  137 

the  resulting  scum  (oxide)  with  the  edge  of  a  card. 
The  reflected  image  of  the  sun  or  star  lies  as  much  below 
the  true  horizon  as  the  real  object  is  above  it,  and  if 
the  angle  between  the  two  is  measured  with  the  sextant 
it  gives  at  once  the  double  altitude  of  the  body,  subject 
to  correction  for  index  error,  etc.  See  §  29  for  an  ex- 
ample. 

60.  Precepts  for  the  Use  of  a  Sextant. — i.  Keep   your 
fingers  off  the  graduation.     It  tarnishes  readily. 

2.  Focus  the  telescope  with  great  care  so  as  to  secure 
sharply  defined  images. 

3.  Make    the    direct    and    reflected    images  equally 
bright,  by  moving  the  telescope  to  or  from  the  plane  of 
the  sextant  with    the  adjusting-screw  provided  for  this 
purpose. 

4.  Bring  the  images  into  contact  midway  between 
the  guide -threads. 

5.  Don't  try  to  hold  the  images  still  in  the  field  of 
view.     Give   the   reflected   image   a   regular   oscillating 
motion  by  twisting  the  wrist,  and  note  its  relation  to  the 
direct  image  as  it  swings  by. 

6.  In  observing  the  sun  take  an  equal  number  of 
observations  on  each  limb  (edge). 

•7.  Take  an  equal  number  of  observations  in  each 
position  of  the  horizon  roof,  direct  and  reversed. 

8.  Determine  the  index  correction  as  carefully  as  the 
angle  which  you  wish  to  measure. 

9.  Whenever  possible  use  a  shade -glass  over  the  eye- 
piece instead  of  those  attached  to  the  sextant  frame. 

10.  Work  as  rapidly  as  you  can  without  hurrying. 


138  FIELD  ASTRONOMY. 

61.  Chronometers. — This  section  will  be  confined  to 
a  consideration  of  the  proper  care  and  use  of  timepieces. 
For  an  account  of  their  mechanical  construction  see  the 
article  Watches  in  the  Encyclopaedia  Britannica. 

A  chronometer  is  a  large  and  finely  constructed 
watch,  whose  face,  hands,  and  train  (wheels)  are  to  be 
considered  as  a  mechanical  device  for  automatically 
counting  and  registering  the  vibrations  of  a  steel  helix, 
called  the  balance-spring.  In  most  chronometers  this 
spring  makes  one  complete  vibration  every  half  second, 
producing  a  beat  (tick)  of  the  chronometer  and  a  forward 
movement  of  the  seconds  hand  through  os.5.  This 
spring  may  vibrate  too  slow  or  too  fast,  thus  producing 
a  rate  of  the  chronometer,  and  it  is  practically  convenient 
that  this  rate  should  be  small,  but  the  real  test  of  excel- 
lence in  a  timepiece  is  not  the  magnitude  of  its  rate,  but 
its  uniformity  of  rate  from  day  to  day. 

In  order  that  the  rate  of  a  chronometer  shall  remain 
constant,  every  precaution  must  be  taken  against  dis- 
turbing the  balance -spring,  and  most  of  the  following 
precepts  for  the  treatment  of  a  chronometer  have  reference 
to  this  condition.  Of  the  various  mechanical  disturbances 
to  which  it  is  subject,  experience  shows  that  a  quick 
rotary  motion  about  the  axis  of  the  balance -spring  is  the 
most  injurious.  According  to  the  chronometer  makers 
a  single  quick  motion  of  this  kind  through  half  a  turn 
and  back  may  change  the  chronometer  correction  several 
seconds  and  so  disturb  the  rate  that  it  will  not  resume 
its  normal  value  for  hours  or  even  days. 

A  chronometer  is  usually  supported  in  gimbals  and 


INSTRUMENTS.  139 

should  be  allowed  to  swing  freely  in  them  when  at  rest, 
in  order  that  it  may  assume  a  vertical  position;  but  when 
carried  about,  the  gimbals  should  be  locked  since  the 
oscillations  that  would  otherwise  be  imparted  to  the 
balance -spring  are  more  injurious  to  the  rate  than  the 
isolated  shocks  that  it  may  receive  when  firmly  held  in 
one  position.  A  chronometer  should  be  kept  in  a  dry 
place,  not  exposed  to  magnetic  influences.  If  possible 
it  should  always  rest  in  the  same  azimuth,  e.g.,  the  zero 
of  the  dial  always  pointing  north.  It  should  be  wound 
at  regular  intervals,  and  its  temperature  should  be  kept 
as  nearly  uniform  as  possible.  The  average  chronometer 
runs  best  at  a  temperature  near  70°  Fahr. 

62.  Comparison  of  Chronometers. — A  problem  of  fre- 
quent recurrence  is  the  comparison  of  one  chronometer 
with  another,  e.g.,  in  order  to  determine  the  correction 
of  one  from  the  known  value  of  AT  for  the  other.  This 
comparison  consists  in  noting  the  time  indicated  by  one 
chronometer  at  a  given  time  shown  by  the  other,  and 
presents  little  difficulty  when  no  greater  accuracy  than 
the  nearest  half -second  is  required.  If  the  comparison 
is  to  be  made  correct  to  the  nearest  os.i,  the  method 
of  coincident  beats  may  be  employed  if  one  of  the  chro- 
nometers keeps  sidereal  and  the  other  solar  time. 

Since  sidereal  time  gains  236  seconds  per  day  uporj 
mean  solar  time  and  the  chronometers  beat  half -seconds, 
there  will  be  472  epochs  during  a  day,  at  which  thfe 
chronometers  beat  in  unison,  i.e.,  a  coincidence  of  the 
beats  occurs  every  three  minutes  throughout  the  day, 
and  if  the  comparison  be  made  at  one  of  these  coinci- 


140  FIELD  ASTRONOMY. 

dences  by  noting  by  each  chronometer  its  indicated 
time  when  the  beats  are  coincident,  no  fractions  of  a 
second  need  be  determined  and  the  comparison  can  be 
made  correct  within  one  or  two  hundredths  of  a  second. 

This  mode  of  comparison  is  illustrated  in  the  follow- 
ing example  of  the  comparison  of  two  mean -time  clocks, 
M  and  F,  with  each  other  by  comparing  each  with  a 
sidereal  clock  designated  H . 

OBSERVED    TIMES    OF    COINCIDENT    BEATS. 

M     iph  30™  49s  F      i9b  34m  55s 

H     10     42       i  H     10     46     o 

The  interval  between  the  coincidences,  as  measured  by 
H*  is  3m  59s  (sidereal),  and  this  interval  reduced  to  mean 
solar  units  and  added  to  M,  or  subtracted  from  F,  gives 
a  comparison  between  the  mean-time  clocks  as  follows : 

M     i9h  34™  473.35  i9h    3om  49*.oo 

F     19     34     55  .00  19     30     56  .65, 

either  form  showing  that  F  was  7s. 6  5  faster  than  M. 

Every  observer  should  acquire  the  ability  to  ' '  carry 
the  beat"  of  a  chronometer,  i.e.,  to  listen  to  and  count 
the  beats  while  attending  to  something  else,  since  nearly 
all  observations  in  which  it  is  required  to  note  the  time 
of  an  event,  e.g.,  the  transit  of  a  star  over  a  thread, 
require  this  ability  unless  special  mechanical  devices, 
such  as  a  chronograph,  are  employed.  (See  §  79.) 


CHAPTER  VIII. 
ACCURATE   DETERMINATIONS. 

63.  General  Principles. — Where  a  high  degree  of 
precision  is  desired  in  the  results  of  observation,  the 
purely  instrumental  sources  of  error  that  have  been 
examined  in  the  preceding  chapter  must  be  eliminated 
by  the  methods  there  shown,  or  by  others  equivalent 
to  them.  But  this  alone  is  not  sufficient,  and  we  note, 
for  example,  that  an  instrument  taken  from  a  warm 
place  and  set  up  in  a  cold  one  undergoes  a  process  of 
cooling  and  contraction  of  its  parts  that,  while  in  prog- 
ress, renders  the  errors  of  adjustment  variable  quantities, 
whose  effects  cannot  be  represented  by  the  formulae 
derived  for  the  case  of  "instrumental  constants."  We 
have  therefore  as  a  rule  to  be  carefully  observed  when 
precision  is  required:  Let  the  instrument  be  set  up  and 
levelled  in  the  place  where  it  is  to  be  used,  at  least  half  an 
hour  before  observations  are  commenced.  Let  the  sur- 
roundings of  the  instrument  during  this  period  be  as 
nearly  as  possible  like  those  under  which  the  observa- 
tions are  to  be  made,  i.e.,  shutters  open,  lamps  lighted, 
etc.  As  a  corollary  to  this  rule  we  have  the  further  pre- 
cept that  during  the  progress  of  the  observations  the 

141 


142  FIELD  ASTRONOMY. 

observer  and  his  lamp  should  be  kept  away  from  the 
instrument  as  much  as  possible. 

There  is  large  room  for  the  display  of  good  judgment 
in  the  selection  of  stars  to  be  observed  for  a  given  pur- 
pose, such  as  the  determination  of  time  or  azimuth,  and 
precepts  bearing  upon  this  choice,  both  with  reference 
to  the  precision  of  the  observations  themselves  and  to 
the  elimination  of  errors  in  the  right  ascensions  and 
declinations  of  the  stars  as  furnished  by  the  almanac,  are 
given  in  the  following  sections. 

Whenever  observations  are  to  be  made  upon  a  con- 
siderable number  of  different  stars,  as  in  determinations 
of  time  and  latitude,  an  observing  list  should  be  prepared 
in  advance,  giving  the  names  and  magnitudes  of  the  stars, 
arranged  in  the  order  in  which  they  are  to  be  observed, 
and  giving  also  such  data  as  may  be  required  for  finding 
them  with  the  given  instrument,  e.g.,  their  right  ascen- 
sions, declinations,  zenith  distances,  etc.  Also,  a  form 
should  be  prepared  in  which  to  record  the  observations, 
each  figure  that  is  to  be  written  down  as  a  part  of  the 
record  having  its  proper  place  allotted  it.  This  place 
must  be  filled  up  before  the  observation  is  complete, 
and  the  presence  of  an  unfilled  space  in  the  form  is  to 
be  considered  as  a  reminder  that  something  remains  to 
be  done. 

64.  Time  by  Equal  Altitudes.  —  The  best  method  of 
determining  time  involves  the  use  of  a  transit  instru- 
ment (see  Chapter  IX),  but  an  excellent  time  determina- 
tion may  be  made  with  a  theodolite,  zenith  telescope, 
or  sextant  by  the  method  of  equal  altitudes,  as  follows : 


ACCURATE  DETERMINATIONS.  143 

We  note  the  chronometer  time,  7\,  at  which  a  star 
west  of  the  meridian  reaxhes  the  zenith  distance  zl  and 
the  time,  T2,  at  which  another  star,  east  of  the  meridian, 
reaches  a  zenith  distance,  z2,  which  differs  as  little  as 
possible  from  zr  In  sextant  observing  it  is  customary 
to  assume  that  if  the  sextant  is  set  to  the  same  reading 
in  the  two  observations  we  shall  have  zl=z2.  For  an 
instrument  of  the  other  type  (theodolite)  the  telescope 
must  be  left  firmly  clamped  in  altitude  as  it  is  turned 
from  one  object  to  the  other,  and  any  slight  change  in 
the  altitude  of  the  line  of  sight  must  be  carefully  deter- 
mined from  readings  of  the  altitude  level  of  the  instru- 
ment. If  the  bubble  changes  its  position  in  the  level- 
tube  when  the  latter  is  turned  from  the  first  to  the  second 
star,  it  should  be  brought  back  to  its  original  place  by 
the  levelling  screws  of  the  instrument,  but  the  angle 
between  the  telescope  and  level-tube  must  not  be  altered. 
If  the  instrument  is  provided  with  an  azimuth  circle,  it 
will  be  well  to  note  its  readings,  ^  and  R2  ,  correspond- 
ing to  the  observed  Tl  and  T2. 

For  the  reduction  of  the  observations  we  take  from 
the  formulae  for  transformation  of  coordinates,  {  14,  the 
equations 

cos  zt  =  sin  0  sin  d1  +  cos  0  cos  d1  cos  ^  , 
cos  z2  =  sin  0  sin  62  ~r  cos  <P  cos  d2  cos  t2  , 

and  in  these  relations  if  we  could  assume, 


we  should  have  at  once, 

cos  *t  =  cos  t2     and     ^  =  —  %. 


144  FIELD  ASTRONOMY. 

From  this  last  relation  we  obtain, 


and  solving  this  for  AT  ,  find, 

^r-iK+oj-i^+r,).  (I04) 

This  ideal  case  may  be  realized  in  practice  by  observing 
the  times  at  which  a  given  star  comes  to  equal  zenith 
distances  on  opposite  sides  of  the  meridian,  i.e.,  before 
and  after  its  culmination,  but  this  may  involve  a  delay 
of  several  hours  between  the  observations,  and  it  will 
usually  be  more  convenient  and  expeditious  to  observe 
in  quick  succession  two  stars  of  nearly  equal  declination 
but  widely  different  right  ascension,  one  east  and  the 
other  west  of  the  meridian,  or  the  sun,  A.M.  and  P.M. 

To  adapt  Equation  104  to  this  case  we  assume  six 
new  quantities,  z,  B,  d,  D,  t,  and  L,  defined  by  the  fol- 
lowing relations: 

z  +  B=zlt     d  +  D  =  dv     t  +  L  =  t1} 
z-B=z2,     d-D  =  d2j     t-L  =  t2) 

and  from  the  last  pair  of  these  equations  we  obtain,  by 
the  method  followed  in  deriving  Equation  104, 

JT^Ka^  +  aJ-KTt  +  TJ+L.  (106) 

To  determine  the  value  of  L  we  introduce  into  Equations 
103  the  quantities  defined  in  Equations  105,  and  sub- 
tracting the  first  of  these  transformed  equations  from 
the  second,  obtain  the  rigorous  relation, 

sin  z  sin  B  =  —  sin  0  cos  d  sin  D 

+  cos  0  sin  d  sin  D  cos  t  cos  L 

•fcos  0  cos  d  cos  D  sin  t  sin  L.       (107) 


.       . 
sin  t          tan  t          cos  0  sin  A 

From  Equations  105  we  find  for  use  here, 


ACCURATE  DETERMINATIONS.  145 

This  equation  is  quite  too  cumbrous  for  use,  but  if  in  the 
plan  and  execution  of  the  observations  care  is  taken  to 
make  B  and  D  small  quantities  whose  cubes  and  higher 
powers  may  be  neglected,  it  is  readily  reduced  to  the 
simpler  form, 

tan  0  _      tan  £  _  B 

(108) 


It  appears  from  these  relations  that  the  quantity  B  is 
half  the  change  of  zenith  distance  suffered  by  the  line 
of  sight  in  passing  from  one  star  to  the  other,  and  this 
change  should  be  measured  with  all  possible  care  by 
means  of  the  altitude  level  of  the  instrument.  If  we 
represent  by  b  the  observed  displacement  of  the  bubble 
between  the  two  observations  and  by  d  the  value  of  half 
a  level  division,  we  shall  have 

B=±bd,  (no) 

where  the  positive  sign  is  to  be  used  when  the  bubble 
stands  nearer  to  the  objective  end  of  the  telescope  at 
the  eastern  than  at  the  western  observation.  The 
value  of  B  is  required  in  seconds  of  time,  and  it  will  there- 
fore be  convenient  to  express  d  in  terms  of  the  same  unit 
instead  of  in  seconds  of  arc. 

The  declination  factor,  D,  should  also  be  expressed 
in  seconds  of  time,  and  since  declinations  are  usually 
given  in  arc,  we  reduce  the  difference  dl  —  d2  to  seconds 


146 


FIELD  ASTRONOMY. 


of  arc,  and  dividing  this  by  15  obtain  in  terms  of  the 
required  unit 


65.  Example.  —  Time  by  Equal  Altitudes.  —  The  follow- 
ing example  illustrates  the  application  of  these  equations 
to  the  reduction  of  observations  made  with  an  engineer's 
transit  provided  with  stadia  threads,  over  which  the 
star's  vertical  transits  were  observed,  the  instrument 
being  turned  between  times  so  that  the  transit  over  the 
horizontal  thread  should  always  occur  near  its  inter- 
section with  the  vertical  thread.  The  three  terms  con- 
tained in  the  value  of  L  (Equation  108)  are  here  repre- 
sented by  the  symbols  Lv  L2,  L3. 

EQUAL    ALTITUDES    FOR   TIME. 

Thursday,  April  30,  1896. 
At  Brick  Pier.     Instrument,  Heyde.     Observer,  C. 


Star 

a  Orionis 

a  Serpentis 

Obs'd  T 

ioh  41'"    9".  6 

ioh  50™  42".  i 

}CTi  +  r2) 

ioh45m   55s-8 

4i      37-2 

5°       14-2 

55-7 

42        5.2 

49      45-3 

55-2 

Level* 

i9-7     3-3 

18.8      2.6 

—  Mean 

-i°    45    55-57 

R 
d 

83°  5i' 
+  7   23   18.8 

277°  23' 
+  6   44   51.3 

L 
i(«i  +  «i) 

+  1       12  .  II 

+  10    44    22.24 

a 

5h  49m  33s-°6 

I5h  39m   ns.  42 

AT 

—  O      21  .  22 

*  The  end  of  the  bubble  nearer  the  objective  is  recorded  first,     d  =  3;/,8  =  o'.zs. 


Ka2—  al) 

4h  54™  49s 

»t-^ 

+  o038'27".5 

bt-b, 

-o.8d 

i(^2~~^l) 

—  4      18 

15 

+  76S_92 

B 

9  .  3010^ 

t  (time) 

4   5°     3i 

tan  $ 

9.9709 

sec  <j> 

o  .  1364 

t  (arc) 

72°  38' 

cosec  t 

o  .0203 

cosec  A 

o  .0032 

a 

7       4 

D 

i.  .8860 

logL3 

9.4406^ 

A.T.            C 

cot  t 

9-4952 

1/70      TJ    \ 
* 

*rO             O 

8-?     14 

—tan  5 

9.09331* 

L, 

+  75-37 

^  o       •*•**• 

logL, 

1.8772 

—     2     98 

logL3 

0.4745** 

L* 

—     0.28 

ACCURATE  DETERMINATIONS.  .147 

For  the  sake  of  illustration  the  reductions  in  the  pre- 
ceding example  are  carried  to  hundredths  of  a  second  of 
time,  but  this  is  a  quantity  quite  inappreciable  in  the 
telescope  of  an  engineer's  transit,  and  with  such  an  instru- 
ment, or  with  a  sextant,  it  will  usually  be  sufficient  to 
carry  the  reductions  to  tenths  of  seconds  only.  Corre- 
sponding to  this  degree  of  accuracy  the  difference  of 
declination  of  the  stars  may  be  as  great  as  two  or  three 
degrees  without  the  introduction  of  sensible  error  into 
the  results  by  reason  of  the  approximate  character  of 
the  reduction  formulag.  The  difference  should  not  ex- 
ceed one  half  of  this  amount  if  hundredths  of  seconds 
are  to  be  taken  into  account. 

66.  Observing  List. — Without  transgressing  these  rather  narrow 
limits  for  dl—  d2,  a  considerable  number  of  suitable  pairs  of  stars  may 
be  selected  from  the  almanac,  as  is  illustrated  by  the  short  observing 
list  given  below,  and  such  a  list  should  be  prepared  for  the  particular 
time  and  place  at  which  observations  are  to  be  made.  At  least  one 
of  the  stars  in  each  pair  should  be  a  bright  one,  easily  recognized  and 
found  with  the  telescope  by  sighting  over  its  tube.  The  second  star 
of  the  pair,  even  though  much  fainter,  may  be  readily  found  by  the 
method  given  below. 

In  the  selection  of  pairs  of  stars  care  should  be  taken  to  secure  those 
that  are  as  near  as  may  be  to  the  prime  vertical  at  the  time  when 
their  altitudes  are  equal,  since  the  motion  in  altitude  is  then  most 
rapid  and  most  accurately  observed.  The  analytical  expression  for 
this  condition  is 

tan  $(£,+£,)  =tan  ^  cos  £(«i  —  <h)',  (II2) 

and  if  this  equation  is  satisfied  by  the  coordinates  of  any  two  stars 
that  differ  but  little  in  declination,  these  stars  will  be  near  the  prime 
vertical  at  the  instant  when  their  altitudes  are  equal.  But  this  con- 
dition should  not  be  too  rieorously  insisted  upon,  and  even  considerable 
deviations  from  it  may  be  permitted  in  order  to  secure  a  suitable 
number  of  bright  stars. 

Having  chosen  a  pair  of  stars,  we  may  determine  as  follows  the 
sidereal  time.  0,  at  which  their  altitudes  will  be  equal:  In  Equation  106 
we  put  JT  =o,  T,  =T2  =0,  and  obtain 

£,  (113) 


148  FIELD  ASTRONOMY. 

where  the  value  of  L  is  to  be  derived  from  Equation  108,  omitting 
the  term  in  B.  It  will  usually  be  convenient  to  observe  the  first  star 
about  five  minutes  before  the  computed  time,  . 

Finding  the  Faint  Star. — If  the  two  stars  have  equal  declinations, 
their  azimuths  at  the  instant  of  equal  altitudes  will  be  numerically 
equal  but  of  opposite  sign,  i.e.,  A1+A2—o,  while  if  their  declinations, 
differ  slightly,  there  will  be  a  small  difference  in  the  azimuths  which 
will  transform  this  equation  into 

To  determine  the  value  of  dA2  in  this  equation  we  obtain  from  the 
astronomical  triangle  the  relation  (Equation  1 5) 

sin  d  =cos  z  sin  0— sin  z  cos  <f>  cos  A,  (us) 

and  differentiating  this,  treating  <f>  and  z  as  constants,  we  find  as  the 
change  of  azimuth  of  the  second  star  produced  by  a  small  change  of 
declination, 

cos  d  dd  =cos  9"  sin  z  sin  A  dA  =cos  9"  cos  d  sin  t  dA ,         (i  16) 
from  which, 

dA= ,    .     4.  (117) 

cos  9  sin  t 

Let  Rlt  R2,  R0  represent  respectively  the  readings  of  the  horizontal 
circle  when  the  telescope  is  directed  to  the  western  star,  to  the  eastern 
star,  and  to  the  meridian;  we  shall  then  have 

A^R.-R,,         A2=R2-R,,  (n8) 

and  substituting  in  Equation  114  these  relations  together  with  the 
approximate  values, 

we  obtain 

Ri+R2  =  2R0^ j-  .2  -,  ,* r.  (120) 

cos  9  sin  i(a2~ai) 

The  last  term  in  this  expression,  computed  for  9"  =43°,  is  tabulated 
in  the  observing  list  under  the  heading  AR,  and  by  means  of  it  and 
the  reading  RI  to  the  first  star  of  a  pair,  the  reading  of  the  horizontal 
circle,  ^2.  mav  be  found  at  which  the  instrument  should  beset  and  the 
arrival  of  the  second  star  in  the  field  awaited  at  a  time  as  much  after 
the  computed  6  as  the  first  observed  time  was  earlier  than  Q.  For 
convenience' sake  orient  the  instrument  and  make  jR0=o.  As  a 
control  upon  the  sign  of  AR,  note  that  the  star  that  has  the  larger 
declination  must  be  the  farther  from  the  south  point. 


ACCURATE  DETERMINATIONS. 


149 


TIME    BY    EQUAL    ALTITUDES. 
Partial  Observing  List  for  <£  =43°. 


Stars. 

Mag. 

R.  A. 

Dec. 

0. 

J*. 

a  Orionis  

O    0 

h.       m. 

C,      co 

0                    / 

7         2  ^ 

h.     m. 

0                / 

a  Serpentis  

2  .  7 

1C         7Q 

6      4< 

10      44 

—  o     54 

a  Can.  Min  
a  Serpentis  

ft  Geminonim  

°-5 
2.3 

I     2 

7      34 
15      39 

7       30 

5     29 
6     45 

28     16 

ii      34 

+  2           0 

fi  Herculis  

•7     e 

17       4"? 

2  7       47 

12      43 

-o     41 

a  Leonis  

I  .  •? 

IO          •? 

12        28 

a  Ophiuchi 

2     2 

1  7       3O 

12         ^8 

13      46 

+  o     16 

ct  Aouilce 

O    O 

1  0       46 

8      ?6 

o  Leonis 

*   8 

9-?6 

I  O        21 

14     44 

-2       28 

67.  Precise  Azimuths.  —  The  azimuth  of  a  terrestrial 
line,  e.g.,  the  line  joining  the  centre  of  a  theodolite  to  a 
distant  mark,  may  be  determined  by  measuring  the 
difference  of  azimuth,  D,  between  the  mark  and  a  star 
at  an  observed  time,  T.  From  the  observed  time  and 
the  right  ascension  of  the  star  its  hour  angle,  t,  may  be 
derived,  and  from  Equations  14  we  obtain,  by  division 
and  introduction  of  the  auxiliary  quantities, 


the  relation 


—  tan  A  = 


gsin 


(121) 


(122) 


I  —  k  COS  t' 

from  which  the  true  azimuth  01  the  star  at  the  time  of 
observation  is  readily  computed.     The  azimuth  of  the 

mark  is  then 

A'=A+D,  (123) 

where  D  is  assumed  to  be  measured  from  the  star  toward 
the  east. 


150  FIELD  ASTRONOMY. 

The  precision  of  A'  depends  equally  upon  D  and  A, 
and  through  A  it  depends  upon  the  assumed  latitude, 
declination,  right  ascension,  and  chronometer  correction 
that  are  employed  in  the  computation.  The  observa- 
tions should  therefore  be  planned  with  reference  to  elim- 
inating whatever  minute  error  may  exist  in  any  of  these 
data,  and  to  overcoming,  by  the  methods  indicated 
below  and  in  §  54,  the  effect  of  instrumental  errors  upon 
the  measured  angle  D. 

Errors  in  the  Assumed  Data. — The  effect  of  these  errors 
may  be  greatly  diminished  by  selecting  for  observation 
a  star  very  near  the  pole  of  the  heavens,  since  the  factor 
g  is  thus  made  small,  and  such  a  star,  e.g.  Polaris,  should 
always  be  chosen.  If  the  chronometer  correction  is  well 
determined,  the  observations  may  be  made  at  any  con- 
venient hour,  whether  near  elongation  or  not.  As  a  guide 
to  the  required  precision  in  JT  we  note  that  for  observa- 
tions of  Polaris  within  the  limits  of  the  United  States 
an  error  of  2s  in  the  time  will  in  no  case  produce  in  the 
computed  azimuth  an  error  greater  than  i". 

If  the  highest  precision  is  required,  the  star  should  be 
observed  at  two  points  of  its  diurnal  path  which  are 
diametrically  opposite  to  each  other,  i.e.,  there  should 
be  two  groups  of  observations  separated  by  an  interval 
of  twelve  hours,  or  some  odd  multiple  of  twelve  hours. 
Errors  in  0,  d,  and  a  will  then  be  almost  perfectly  elim- 
inated, and  there  will  also  be  eliminated  any  systematic 
personal  error  of  observation  depending  upon  the  direc- 
tion of  the  star's  apparent  motion,  such  as  is  sometimes 
found  to  exist  in  the  work  of  even  the  best  observers. 


ACCURATE  DETERMINATIONS.  151 

A  similar  but  less  complete  elimination  of  errors  may  be 
obtained  from  observations  made  at  a  single  epoch  if 
these  are  equally  divided  between  stars  upon  opposite 
sides  of  the  pole  and  equidistant  from  it.  Examples  of 
pairs  of  stars  which  approximately  fulfil  this  condition 
are  Polaris  and  6  Ursae  Minoris;  51  H.  Cephei  and 
d  Ursae  Minoris. 

The  angle  D  may  be  measured  with  either  a  repeating: 
or  a  non-repeating  (direction)  instrument,  and  the  student 
should  observe  the  following  respects  in  which  their  use 
differs:  For  a  repeating  instrument  the  azimuth  level 
should  be  used  to  determine  the  inclination  of  the  verti- 
cal axis  corresponding  to  the  lower  motion  of  the  instru- 
ment. For  a  non-repeating  instrument  the  inclination 
to  be  determined  is  that  of  the  horizontal  axis.  In  both 
cases  the  bubble  readings  are  to  be  taken  when  the  line 
of  sight  is  directed  toward  the  star  and  also  when  it  is 
turned  toward  the  mark,  unless  the  latter  has  a  zenith 
distance  of  90°,  in  which  case  erroneous  levelling  will  not 
affect  the  readings  to  it. 

With  any  type  of  instrument  the  horizontal  circle 
is  to  be  turned  in  its  own  plane  from  time  to  time  during 
the  observations,  so  that  the  vernier  or  microscope  read- 
ings shall  be  symmetrically  distributed  throughout  the 
entire  360°  of  the  graduation;  e.g.,  for  an  instrument  with 
two  microscopes  let  one  ninth  of  the  total  number  of 
observations  be  made  with  the  circle  reading  to  the 
mark  approximately  o°,  another  ninth  with  the  circle 
reading  20°,  40°,  60°,  etc.  But  see  §  53  for  the  peculiar 


152  FIELD  ASTRONOMY. 

manner  in  which  the  circle  settings  should  be  changed 
in  the  case  of  a  repeating  instrument. 

Level  Corrections. — The  correction  for  level  error  is 
to  be  applied  to  each  circle  reading  as  shown  in  §  50, 
but  for  observations  made  by  the  method  of  repetitions 
the  level  correction,  b'  tan  h,  there  given  for  the  reading 
to  the  star,  must  be  multiplied  by  n,  the  number  of  point- 
ings in  a  set,  since  the  difference  of  the  corrected  read- 
ings to  star  and  mark  is  to  be  divided  by  n  in  order  to 
obtain  the  measured  angle.  It  will  usually  be  expedient 
to  arrange  the  form  of  record  of  the  observations  so  that 
the  level  corrections  may  be  applied  and  the  angles  worked 
out  in  the  record  book. 

68.  Reduction  of  the  Observations.  —  After  the  hour 
angles  have  been  formed  from  the  relation  t  =  T  +  4T  —  a, 
and  the  constants  g  and  k  computed  with  the  known  decli- 
nation and  latitude,  the  computation  of  A  presents  no 
difficulties,  but  it  may  be  considerably  abbreviated 
through  the  use  of  Albrecht's  Tables  (reproduced  in 
Appendix  VII,  Annual  Report  U.  S.  Coast  and  Geodetic 
Survey,  1897-98),  which  with  the  argument  log  x  give 

the  logarithm  of  — - — .     Calling  this  last  factor  F,  and 
i      x 

putting  k  cos  t  =  x,  Equation  122  assumes  the  very 
simple  form 

—  tan  A=gFsin*.  (124) 

In  the  absence  of  special  tables  for  F  its  value  may 
be  readily  obtained  from  an  ordinary  table  of  addition 
and  subtraction  logarithms  as  follows:  Representing 


ACCURATE  DETERMINATIONS.  153 

by  A  and  B  the  argument  and  function  in  such  a  table,* 
i.e.,  A=logx,  B  =  log(i+x),  we  have,  whenever  cos  t 
is  negative,  A  =  log  (k  cos  t),  log  F=  —B.  When  cos  t 
is  positive,  we  use  the  development 


,  etc., 


and  interpolating  from  the  table  of  addition  logarithms 
the  values  of  Bv  B2,  Bv  etc.,  corresponding  to  the  argu- 
ments x,  x2,  x4,  we  find 

logF  =  Bl  +  B2  +  B4  +  etc.  (125) 

For  observations  of  Polaris  made  within  the  limits  of  the  United 
States  it  will  never  be  necessary  to  use  more  than  the  first  two  terms 
of  this  series,  e.g.,  corresponding  to  this  case  the  greatest  possible 
value  of  k  cos  t  furnishes  log  x  and  the  several  values  of  B  given  below: 

log*     8.39386     BI  0.0106248 
log*2    6.7877       £2  2663 

log*4    3-575         #4  2 

logF  0.0108913 

In  ordinary  practice  the  value  of  log  F  will  be  required  to  only  six 
places  of  decimals,  and  B^  +B2  furnishes  this  degree  of  precision, 

Where  the  highest  degree  of  precision  is  sought,  it  is 
customary  in  the  reduction  of  the  observations  to  com" 
pute  for  each  observed  time  the  corresponding  value  ol 
A,  but  this  process  may  be  very  greatly  abridged  by 
treating  the  mean  of  a  considerable  number  of  observa- 
tions as  a  single  observation  made  at  T0,  the  mean  of  the 
recorded  times.  The  azimuth  A0  computed  from  T0 
will  not  correspond  exactly  to  the  observations,  but  the 
correction  required  on  this  account  is  readily  obtained. 

*  Do  not  confound  this  use  of  A  with  its  wholly  different  meaning 
in  Equation  124. 


154  FIELD  ASTRONOMY. 

We  may  develop  by  Taylor's  Formula  the  relation  be* 
tween  azimuth  and  time  in  the  form, 

A  -A+ 


an  equation  which  obtains  for  each  observed  T  and  its 
corresponding  A.  If  we  take  the  mean  of  these  several 
equations  and  note  that  the  mean  of  the  (T—T0)s  is 
necessarily  zero,  since  T0  is  the  mean  of  the  Ts,  we  find 
for  the  average  of  the  set, 

where  the  last  term  of  the  expression  is  the  required 
correction  to  reduce  AQ  to  the  mean  of  the  observed 
azimuths.  For  the  numerical  application  of  this  formula 
we  need  to  introduce  a  convenient  expression  for  /"(A0), 
and  there  must  also  be  a  numerical  factor  such  that  the 
value  of  the  term  shall  be  given  in  seconds  of  arc  when 
T—T0  is  expressed  in  minutes  of  time.  This  factor, 
combined  with  the  coefficient  J  which  appears  in  the 
equation,  is  readily  shown  to  be, 

/6oXi5\*     206265 

r^7f7    x—  r-2-  =*  [0-2930]. 


The  differential  coefficient,  f"(AQ),  does  not  admit 
of  an  expression  that  is  both  simple  and  rigorous,*  but, 
with  entire  accuracy  at  the  pole  and  approximately  for 
any  star  near  the  pole,  we  may  write 
/"(A0)=sinA0, 

*  See  p.  207  for  the  complete  expression  for/"  (A0). 


ACCURATE  DETERMINATIONS.  155 

and  combining  these  several  expressions  we  find  as  the 
correction  to  the  computed  azimuth,  A0, 

4A0  =+[0.2930]  sin  A^Z(T-TQYt          (127) 

where  n  is  the  number  of  Ts  included  in  the  mean,  Tm 
and  the  differences,  T  —  T0,  are  to  be  expressed  in  minutes 
of  time.  AAQ  must  always  be  so  applied  as  to  bring 
the  computed  A0  nearer  to  the  meridian. 

See    §  85    for    the  extremely  small  effect  of  diurnal 
aberration  upon  azimuth  determinations. 

69.  Example.  —  Precise  Azimuth.  —  The  example   on 
p.  156  represents  a  determination  of  azimuth  made  with 
an  engineer's  transit,  using  the  method  of  repetitions, 
four  pointings  in  a  set,  and  combining  two  sets  in  such 
a  way  as  to  eliminate  the  effect  of  lack  of  parallelism  of 
the  axes  of  the  instrument,  see   §  53.     The  graduation 
errors  are  not  here  eliminated,  and  other  sets  with  read- 
ings symmetrically  distributed  about  the  circle  are  re- 
quired for  this  purpose. 

70.  Precise  Latitudes.  —  Zenith  Telescope  Method. — In 
Fig.  12  let    V  represent  any  point  on  the  meridian,  5^ 


POLE 

FIG.  12.— Zenith  Telescope  Latitudes. 

and  52  the  points,  on  opposite  sides  of  V,  at  which  two 
stars,  of  declination  d1  and  d2  respectively,  cross  the 
meridian  in  their  diurnal  motion,  and  let  zl  and  z2  denote 


156 


FIELD  ASTRONOMY. 


PRECISE   AZIMUTH    DETERMINATION. 

At  Station  M.     Monday,  May  i,  1899. 
Instrument  No.  386.     Chronometer,  S.     Observer,  C. 
Chronometer  JT '  =  — 2'n39s.7.     4  tan  h.d  =  io".'j. 


Object. 

Circle 
n. 

L 
L 

4 
R 

R 

4 

Chronometer. 

Horizontal  Circle. 

Vert.  Circle. 
Levels. 

Angle. 

Ver.  A. 

Ver.  B. 

Mark  
Polaris.  . 

Polaris  .  . 
Mark.  .  .  . 

h.     m.     s. 
12    25    .. 

12     27     28 

30  59 
33   28 
36   24 

o        /      // 
150    31    25 

117    21    40 
117    22       O 

150   10     o 

/      // 
31    15 

21   35 

21     30 

9  40 

0             / 

O       O 
W.          E. 

8.5     15.0 

12.7    n  .0 

o          /       // 
150    31     20 

=               -26 
117     21     38 

—  2.4 

4i      5° 

4i      43 
6.3      17.4 
18.0        5.8 

)i28   19 

)33    10     8 

12     32        4.8 

12  40     7 
45   28 
47   38 
49  40 

8   17   32 
117   21   45 

+  6 
!5°     9  5° 

+  Q-55 

0       0 

12    52    .. 

)i82   53 

)32   47   59 

12  45  43.2 

8     12        0 

REDUCTION. 

h.    m.      s. 

h.    m.     s. 

t 

a 
sec  <f> 
cotd 
tan  <£ 

43     4  37 
88  46   12.5 

I     21     28.O 

o.  13641 
8.33180 
9  .97082 

T+AT-a 
t 
cos  t 
k  cos  t 
sin  t 

12     29    25.1 

11      7   5  7  •  i 
i66°59'i6" 
9  .  98870^ 
8  .  29132*1. 
9-35249 

12   43      3-5 

11    21    35.5 
1  70°  23'  52" 
9  .  99388^ 
8  .  2965011 

9  .  22221 

8.46821 

g 

8.46821 

8.46821 

8.30262 

F--B, 

9-99I59 

9.99149 

—  tan  A0 

7  .81229 

7  .68191 

(T  —  TO) 

20,  1,2,  18     -o,  o, 

4,  16          A0 

179  37  4i 

179  43   28 

[2(T  —  T0) 

i  .009              i  .  CK 

)7              4A0 

o 

o 

sin  AQ 

7.812^          7.6* 

\2K 

Const. 

o.293tt          o.2( 

>3«            D 

8   17   32 

8     12        0 

logJA0 

9.114             9.o; 

12 

A  A, 

+  o".i3            +o' 

'.12                  A' 

187  55  13 

187   55   28 

The  corrections  AA^  computed  above  are  too  small  to  be  taken 
into  account  in  observations  of  this  character,  but  with  a  larger  instru- 
ment or  when  the  star  is  near  elongation  they  become  of  sensible 
magnitude. 


ACCURATE  DETERMINATIONS.  157 

the  arcs  VS1  and  VS2.     Denoting  by  0"  the  declination 
of  V,  we  have  from  the  figure 


and  by  subtraction, 

+  01-*2).  (128) 


Since  V,  by  supposition,  is  any  point  of  the  meridian, 
we  may  now  define  it  as  the  projection  upon  the  merid- 
ian, of  the  point  in  which  the  vertical  axis  of  a  theod- 
olite or  other  similar  instrument  meets  the  celestial 
sphere,  and  we  may  represent  by  b"  the  zenith  distance 
of  V,  reckoned  positive  when  the  zenith  lies  between  V 
and  the  pole.  Since  the  latitude  is  equal  to  the  declina- 
tion of  the  zenith,  we  shall  have 


In  the  practice  of  American  government  surveys 
all  precise  determinations  of  latitude  are  based  upon 
this  equation  and  are  made  with  an  instrument,  the 
zenith  telescope,  especially  designed  for  the  micrometric 
measurement  of  small  differences  of  zenith  distance, 
the  zt  —  z2  of  the  equation.  But  Equation  129  may  be 
applied  with  any  instrument  capable  of  measuring  alti- 
tudes —  theodolite,  sextant,  etc.  —  and  in  general  it  will 
furnish  better  results  than  other  modes  of  using  the 
instrument,  since  if  the  stars  are  so  selected  that  zl 
differs  but  little  from  z2,  any  constant  errors  which  may 
be  present  in  the  instrumental  work  will  be  very  nearly 
the  same  for  the  two  stars,  and  will  be  approximately 
eliminated  from  the  difference  zl  —  z2.  We  shall  here 


158  FIELD  ASTRONOMY. 

develop  the  zenith-telescope  method  with  reference  to 
its  use  with  an  engineer's  transit  provided  with  a  gradi- 
enter  and  an  altitude  level,  which  latter  may  be  its 
striding-level  properly  fastened  to  the  alidade  at  right 
angles  to  the  horizontal  axis.  With  very  small  modi- 
fications the  resulting  formulas  will  be  applicable  to  the 
zenith  telescope  as  usually  constructed. 

The  first  step  in  the  application  of  the  method  is  the 
selection  of  an  observing  programme,  consisting  of  a 
number  of  pairs  of  stars  whose  right  ascensions  and 
declinations,  for  each  pair,  satisfy  the  conditions 


a2-a1<2om,          2  +   1-20<±G,  (130) 

where  G  denotes  the  greatest  angle  that  can  be  con- 
veniently measured  with  the  gradienter.  Write  upon 
the  edge  of  a  slip  of  paper  the  approximate  value  of  20, 
and  turning  to  a  suitable  list  of  stars,  e.g.,  the  list  of 
mean  places  given  in  the  almanac,  subtract  each  decli- 
nation in  turn  from  2  0  and  seek  within  the  given  limits 
of  right  ascension  a  star  whose  declination  differs  but 
little  from  the  difference  thus  obtained.  If  bright  enough 
to  be  observed  with  the  given  instrument,  any  two  stars 
thus  related  will  constitute  a  latitude  pair. 

Having  prepared  such  an  observing  list,  before  the 
first  of  these  stars  comes  to  the  meridian  let  the  instru- 
ment be  carefully  levelled  and  oriented  and  its  telescope 
set  to  the  approximate  zenith  distance  of  the  star, 
zl  =  ±(^—dl).  When  the  star  by  its  diurnal  motion 
is  brought  into  the  field  and  passes  behind  the  vertical 
thread,  a  pointing  in  altitude  should  be  made  upon  it 


PLATE  V. 


A  Zenith  Telescope  as  used  at  the  International  Latitude  Stations.     Length  of 
Telescope  52  inches.     Approximate  Cost  $1600. 

[To  face  p.  158.] 


ACCURATE  DETERMINATIONS,  159 

with  the  gradienter,  and  the  readings  of  the  altitude 
level  and  gradienter  head  recorded  immediately  after 
the  pointing.  Leaving  the  telescope  firmly  clamped 
in  altitude,  let  it  be  now  revolved  180°  in  azimuth  with- 
out loosing  the  altitude  clamp,  and  with  the  gradienter 
bring  the  line  of  sight  to  the  zenith  distance  of  the  second 
star,  z2=  T  (0  —  <^2)>  and  observe  it  precisely  as  before. 
If  the  level-bubble  changes  its  position  in  the  tube  as  the 
instrument  is  turned  from  the  first  to  the  second  star,  it 
should  be  brought  back  to  its  initial  position  by  means 
of  the  levelling  screws. 

The  readings  of  the  level  in  the  two  positions  deter- 
mine the  average  value  of  b" ,  and  if  Rl  and  R2  represent 
the  respective  gradienter  readings  and  k  is  the  angle 
moved  over  by  the  line  of  sight  when  the  gradienter  is 
turned  through  one  complete  revolution,  we  shall  have, 

z1-z2=±k(Rl-R2).  (131) 

71.  Minor  Corrections. — Before  introducing  this  value 
into  the  expression  for  20  we  proceed  to  examine  some 
matters  that  require  further  explanation,  viz. : 

Level  Error. — The  small  term  26"  arises  from  a  devia- 
tion of  the  vertical  axis  of  the  instrument  from  the  true 
vertical  Its  amount  and  sign  are  to  be  determined 
from  readings  of  an  altitude  level,  as  shown  in  §  42, 
Make  this  error  small  by  turning  the  levelling  screws, 
if  necessary,  so  that  the  bubble  readings  shall  be  the  same 
for  the  second  star  as  for  the  first. 

Refraction. — The  effect  of  refraction  upon  the  latitude 
observations  is  most  readily  determined  by  substituting, 


160 


FIELD  ASTRONOMY. 


in  place  of  the  true  declinations  of  the  stars,  their  appar- 
ent declinations  as  affected  by  the  refraction.  This  dis- 
places each  star  toward  the  zenith  by  the  amount,  (§23) 

982"  B 


tan  z\ 


(132) 


and  since  for  the  southern  star  this  displacement  in- 
creases, while  for  the  northern  star  it  diminishes,  the 
declination,  we  shall  have  as  the  sum  of  the  apparent 
declinations, 

(tan  z,  -  tan  zt)  , 


which  is  equivalent  to, 
982 


[ 


B 


cos  zl  cos  z2     456  +  tj 


«0°.    (133) 


The  following  table  gives  the  value  of  the  bracketed  coefficient  in 
this  equation,  computed  with  the  argument  z=^(zl+z2),  for  an  aver- 
age condition  of  the  atmosphere,  barometer  29.00  inches,  temperature 
50°  Fahr.  In  all  ordinary  cases  the  correction  for  refraction  may 
be  found  with  sufficient  accuracy  by  multiplying  the  tabular  number, 
5,  by  the  difference  of  the  zenith  distances  of  the  two  stars,  expressed 

in  degrees, 

r=s(zl-z2)°.  (134) 

Since  5  is  a  positive  number,  the  correction  thus  found  will  always 
have  the  same  sign  as  the  term  zl—  z2,  measured  with  the  gradienter. 

REFRACTION    COEFFICIENTS. 


s 

j 

z 

j 

0° 
IO 

20 

3° 
40 

•••  - 

T'?     4 

'     7 

50° 

55 
60 

65 

70 

3-'o4"   6 
3-9   J 

84   29 

5° 

2.4    7 

75 

14-5 

The  use  of  the  table  is  illustrated  in  the  following  short  example 
taken  from  the  data  of  §  73: 


z 

Zl-Z2 

s 
r 


5°-9 
5-6 

'"•5 
14.0 


ACCURATE  DETERMINATIONS.  161 

Reduction  to  the  Meridian. — It  is  sometimes  conve- 
nient or  necessary  to  observe  a  star  at  some  other  instant 
than  that  of  its  meridian  passage,  and  for  this  purpose 
the  instrument  may  be  turned  out  of  the  meridian,  set 
at  an  azimuth  that  we  will  represent  by  a',  and  the  ob- 
servation made  precisely  as  before.  It  is  evident  that 
this  is  equivalent  to  observing  on  the  meridian  a  star 
whose  meridian  altitude  is  equal  to  the  altitude  of  the 
given  star  at  the  moment  of  observation,  and  whose 
declination,  therefore,  differs  from  that  of  the  latter  star 
by  the  reduction  to  the  meridian  corresponding  to  the 
azimuth  a',  (Equation  55).  In  the  reduction  of  the  ob- 
servation we  have  therefore  to  substitute  in  place  of 
the  star's  true  declination,  d,  a  corrected  declination,  d", 
given  by  the  relation 

d"  =  d±f(a')*,      /  =  [7-9407]  cos  0  cos  h0  sec  d,     (135) 

where  a'  is  to  be  expressed  in  minutes  of  arc  and  the 
upper  sign  applies  to  a  star  between  the  zenith  and  pole, 
the  lower  sign  to  all  other  cases. 

For  the  sake  of  increased  precision  it  will  frequently 
be  advantageous  to  make  several  gradienter  pointings 
upon  a  star  in  different  azimuths,  during  the  two  or 
three  minutes  nat  precede  and  follow  its  culmination, 
and,  having  first  oriented  the  instiumciit,  to  determine 
from  readings  of  the  horizontal  circle  the  corresponding 
azimuths  required  in  the  reduction. 

72.  Errors  of  the  Screw.  —  In  Equation  131  it  is 
tacitly  assumed  that  the  angle  moved  over  by  the  line 
of  sight  when  the  gradienter  is  turned  from  one  reading 


162  FIELD  ASTRONOMY. 

to  another  is  strictly  proportional  to  the  amount  of 
turning  of  the  screw.  This  is,  however,  an  ideal  con- 
dition seldom  realized  in  fact,  and  if  the  capabilities 
of  the  instrument  are  to  be  fully  utilized  the  errors  of 
the  gradienter  must  be  investigated,  and  a  set  of  cor- 
rections, C,  determined,  such  that  the  angle  moved 
through  by  the  line  of  sight  when  the  gradienter  is  turned 
from  the  reading  R^  to  R2  may  be  strictly  proportional 
to  the  difference  of  the  corrected  readings,  R  = 


R").  (137) 

This  calibration  of  the  screw  may  be  made  as  follows  : 
Let  some  fixed  vertical  angle,  e.g.,  the  difference  of  ele- 
vation of  two  terrestrial  points,  be  measured  upon  con- 
secutive parts  of  the  gradienter  screw,  from  the  begin- 
ning to  the  end  of  its  run,  so  that,  calling  this  angle  v, 
we  shall  have, 


(137*) 


The  second  reading  of  the  screw  in  the  first  measurement 
of  v  must  be  the  same  as  the  first  reading  in  the  second 
measurement,  etc.,  and  to  secure  this  the  gradienter 
should  not  be  touched  after  the  second  pointing,  Rlt 
has  been  made,  but  the  telescope  should  be  undamped, 
set  back  by  hand,  approximately,  upon  the  first  point 


ACCURATE  DETERMINATIONS.  163 

and  the  accurate  pointing  completed  by  means  of  the 
levelling  screws. 

From^the  mean  of  the  preceding  equations  we  obtain 

i-f^v^. 

which  contains  the  three  arbitrary  quantities  k,  Cm,  C0, 
and  is  the  only  equation  that  these  quantities  are  re- 
quired to  satisfy.  We  may  therefore  impose  two  addi- 
tional relations  among  them,  and,  as  convenient  ones  for 
the  present  purpose,  we  assume  Cm  =  C0  =  c,  where  c  is 
a  constant  whose  value  we  shall,  for  the  present,  leave 

ni 

undetermined.     Representing  by  p  the  value  of  T   cor- 

K 

responding  to  these  assumptions, 


and  introducing  it  into  Equations  137,  we  find  the  fol- 
lowing results  : 

C0=  +c 

CX#0  +  p)  -R,  +c, 

C2  =  (R0  +  2p)  -R2  +c, 

R*+c,  (140) 


The  corrections  thus  derived  from  the  readings,  R, 
may  be  plotted  in  a  curve,  from  which  values  of  C  for 
all  intermediate  readings  may  be  obtained.  The  par- 
ticular value  assigned  to  c  will  have  no  influence  upon 


164 


FIELD  ASTRONOMY. 


the  shape  of  this  curve,  but  will  determine  its  position 
with  respect  to  the  axis  of  x,  and  we  may  assign  to  c 
with  advantage  a  value  that  will  make  the  entire  curve 
lie  above  the  #-axis,  i.e.,  one  that  will  make  all  the  values 
of  C  positive  quantities. 

The  following  example  represents  the  record  and 
reduction  of  a  set  of  readings  made  for  the  investigation 
of  the  errors  of  the  gradienter  of  an  engineer's  transit. 
The  quantities  in  the  column  R  are  those  directly  ob- 
served; the  column  m  gives  the  serial  number  corre- 
sponding to  that  used  in  the  above  analysis. 

Thursday,  June  7,  1900. 
Gradienter  of  Instrument  No.  386.     Observer,  P. 


•m 

R 

/?„+«// 

(R^+ntfi-R 

c 

0 

o  .027 

0.027 

0.000 

+31 

I 

2.037 

2  .025 

—  .012 

19 

2 

4.045 

4.023 

—  .022 

9 

3 

6.047 

6  .022 

-.025 

6 

4 

8.048 

8  .020 

—  .028 

3 

5 

10  .049 

10  .018 

-.031 

0 

6 

12.047 

12  .Ol6 

-.031 

0 

7 

14-045 

14.014 

—  .031 

0 

8 

16.035 

16.013 

—  .022 

9 

9 

18  .022 

18  .on 

—  .on 

20 

10 

20  .009 

20.009 

0  .  000 

+  3i 

In  the  reduction  of  the  above  we  use 

^  =  ^0(20.009  —  0.027)  =  1.9982. 

The  last  column,  expressed  in  thousandths  of  a  revolu- 
tion, is  obtained  by  adding  to  the  numbers  of  the  pre- 
ceding column  the  assumed  constant,  c—  +0.031.  A 
considerable  number  of  such  determinations  should  be 
made  and  the  mean  of  the  several  results  adopted  as 
definitive  corrections  to  the  gradienter  readings.  Similar 


ACCURATE  DETERMINATIONS.  165 

corrections  must  always  be  applied  where  a  high  degree 
of  precision  is  required  in  the  use  of  a  gradienter  or 
other  similar  micrometer,  e.g.,  the  eyepiece  micrometer 
of  a  zenith  telescope  or  transit,  and  particular  care  should 
be  given  to  them  in  determinations  of  k,  the  value  of  one 
revolution  of  the  gradienter  screw. 

In  a  similar  manner  the  gradienter  should  be  exam- 
ined for  periodic  errors,  i.e.,  errors  peculiar  to  a  particular 
part  of  a  turn  and  which  repeat  themselves  whenever 
the  same  part  of  the  head,  as  the  o,  comes  under  the 
index,  regardless  of  the  number  of  whole  revolutions 
at  which  the  screw  stands. 

73.  Gradienter  Latitudes.  Example.  —  We  may  now 
write  the  equation  for  zenith-telescope  latitudes  in  the 
form, 

(141) 


through  which  a  value  of  the  latitude  may  be  derived 
from  each  pair  of  stars  observed,  if  k  is  known.  This 
value  of  a  revolution  of  the  screw,  k,  may  be  determined 
by  measuring  with  the  gradienter  a  known  angle,  such 
as  the  difference  of  declination  of  two  stars,  or  it  may 
be  treated  as  an  unknown  quantity  whose  value  is  to 
be  derived  from  the  latitude  observations  themselves. 
In  the  latter  case  at  least  two  pairs  of  stars,  preferably 
ten  to  twenty  pairs,  must  be  observed  for  the  deter- 
mination of  the  two  unknowns,  0  and  k,  and  these  should 
be  so  selected  that  in  one  pair  the  sum  of  the  declinations 
is  greater  than  20  and  in  the  other  pair  is  less  than  20. 
The  following  example  represents  the  observation 


166  FIELD  ASTRONOMY. 

and  reduction  of  a  single  pair  of  stars  made  with  the 
instrument  shown  in  Plate  I,  whose  errors  are  investi- 
gated in  §  72.  The  gradienter  readings  as  directly 
observed  are  given  in  the  column  marked  R,  and  in  the 
following  column  there  are  given  the  corrections  to 
these  readings  as  interpolated  from  the  table  at  p.  164. 
The  instrument  having  been  oriented  by  the  method 
of  §  32,  the  readings  of  the  horizontal  circle,  in  the  column 
H.C.,  furnish  immediately  the  azimuths,  a' ,  required 
for  computation  of  the  reductions  to  the  meridian,  which 
are  here  represented  by  the  letter  M.  The  stars'  merid- 
ian altitudes,  h0,  that  are  also  needed  for  the  compu- 
tation of  these  reductions,  may  be  obtained  with  suffi- 
cient accuracy  from  the  declinations  and  the  known 
approximate  latitude  of  the  place,  43°.  The  value  of 
a  revolution  of  the  gradienter,  k,  was  known  to  be  about 
20'  30",  and  this  value  together  with  the  observed  dif- 
ference of  the  gradienter  readings  determines  zl  —  z2 
with  sufficient  precision  to  permit  the  refraction  correction 
to  be  interpolated  from  the  table  at  p.  160.  The  level 
correction,  26"  =  —  7",  is  negative  since  the  level  read- 
ings show  that  the  vertical  axis  of  the  instrument  pointed 
north  of  the  zenith,  i.e.,  in  too  great  a  latitude. 

The  declinations  of  the  stars  are  taken  from  the 
American  Ephemeris,  but  in  the  case  of  Polaris,  which 
was  observed  at  its  transit  over  the  lower  half  of  the 
meridian,  sub  polo,  the  almanac  declination  is  subtracted 
from  1 80°  in  order  to  obtain  the  distance  of  the  star 
from  the  upper  half  of  the  equator,  which  is  the  quantity 
used  in  the  analysis  and  required  in  the  reduction. 


ACCURATE  DETERMINATIONS. 


167 


Monday,  May  20,  1901. 
At  Azimuth  Stake.     Instrument  No.  389.     Observer,  C. 


Star. 

Level. 

N.           S. 

R 

Corr. 

H.  C. 

s 

M 

Remarks. 

o         / 

o       i      n 

it 

Polaris,  S.P. 
a  Virginis 

6.2   9.2 
8.0  7.5 

0.302 

16  .  704 

+  29 

+  13 

J79  55 
358  21 

91    13   18 
-io  38   57 

-6 

-51 

Level,  d  =  3" 

-1.25 

Reduction  to  Meridian. 

Star 

Polaris 

a  Virginis 

VtV 

a' 

5' 

99' 

cos  <f> 

9.863 

9.863 

26" 

COS    UQ 

9.872 

9.906 

sec  d 

i  .671 

o  .008 

Ref'n 

Const. 

7.941 

7.941 

(a')2 

1.398 

3-992 

1?'  -/?" 

logM 

°-745 

1.710 

M 

-5-6 

-51.2 

2<£  =  8o°33'3 

8o°  33'  24" 

~7     ! 
+  14    i 

1 6 .  386  rev. 
)'  3i"  +  i6.386fe 

Each  observed  pair  of  stars  furnishes  an  equation 
similar  to  the  above,  involving  0  and  k  as  unknown 
quantities,  for  which  definitive  values  are  to  be  obtained 
from  a  solution  of  all  the  available  equations.  For  illus- 
tration we  select  a  single  additional  pair  of  stars  and  its 
resulting  equation,  viz., 

20  =  91°  9'  52"—  14.671^, 
and  combining  it  with  the  one  derived  above  we  obtain, 

&  =  i229".4,  0  =  43°  4'  38"- 

This  value  of  0  agrees  within  i"  with  the  known  latitude 
of  the  place  of  observation  and  represents  about  the 
limit  of  accuracy  attainable  with  an  engineer's  transit. 

With  the  zenith  telescope,  used  in  essentially  the 
same  manner  as  above,  a  precision  of  about  o".i  is 
attained.  See  Appendix  7,  Report  of  the  U.  S.  Coast 
and  Geodetic  Survey  for  the  Year  1897-98,  for  an  exposi- 
tion of  the  methods  employed  with  such  an  instrument. 


CHAPTER  IX. 
THE   TRANSIT   INSTRUMENT. 

74.  General  Principles.  —  Adjustments  of  the  Instru- 
ment.— If  the  celestial  meridian  were  a  visible  line  drawn 
across  the  heavens,  the  local  sidereal  time  corresponding 
to  this  meridian  might  be  determined  by  observing  the 
chronometer  time,  T,  at  which  a  star  of  known  right 
ascension,  a,  crossed  this  line.  We  should  then  have 
for  the  correction  of  the  timepiece  employed, 

AT  =  a-T. 

The  transit  instrument,  different  forms  of  which  are  shown 
in  the  Frontispiece  and  in  Plate  VI,  is  a  substitute  for 
the  visible  meridian  above  supposed.  Its  essential  parts 
are  illustrated  by  the  telescope  and  standards  of  a  large 
theodolite  firmly  mounted,  with  the  horizontal  axis  of  ro- 
tation perpendicular  to  the  plane  of  the  meridian,  i.e.,  east 
and  west,  and  level.  The  telescope  is  usually  provided 
with  several  vertical  threads  (an  odd  number  of  them), 
each  of  which,  as  seen  by  the  observer,  is  projected  against 
the  sky  as  a  background,  and  each  of  which,  when  the 
telescope  is  turned  about  the  rotation  axis,  traces  upon 
the  sky,  by  virtue  of  this  rotation,  a  circle  whose  plane  is 
perpendicular  to  the  axis.  Also,  one  or  more  horizontal 
threads  are  usually  introduced  to  mark  the  middle  points 

of  the  transit  threads. 

168 


PLATE  VI. 


A  Straight  Transit   Instrument.     Length   of  Telescope   30  inches. 
Approximate  Cost  $1000. 

\Tofacep.  168.] 


THE   TRANSIT  INSTRUMENT.  169 

A  transit  instrument  is  said  to  be  perfectly  adjusted 
when  the  circle  thus  traced  upon  the  sky  by  its  middle 
vertical  thread  coincides  with  the  local  meridian,  and  for 
such  an  instrument  it  is  evident  that  the  time  of  a  star's 
transit  over  this  thread  may  be  substituted  for  the  time 
of  its  transit  over  the  visible  meridian  above  supposed, 
and  the  chronometer  correction,  AT,  will  then  be  fur- 
nished by  the  equation  printed  above.  But  in  general 
it  cannot  be  assumed  that  these  adjustments  are  perfect, 
and  we  must  consider  them  as  so  many  possible  sources 
of  error  whose  effects  must  be  in  some  way  eliminated 
from  the  results  of  observation. 

Optical  Adjustments. — We  assume  that  great  care 
has  been  given  to  the  optical  adjustment  of  the  instru- 
ment, so  that  both  the  transit  threads  and  the  star  are 
sharply  defined  and  distinctly  seen.  For  this  purpose 
the  eyepiece  should  first  be  so  set  that  the  threads  appear 
black  and  distinct,  and  threads  and  eyepiece  should 
then  be  moved  in  or  out  together  until  a  star,  preferably 
a  double  star,  presents  a  clear  image  without  trace  of 
fuzziness,  projecting  rays,  or  stray  light.  This  last  ad- 
justment may  be  a  little  more  accurately  made  by  cov- 
ering the  upper  half  of  the  telescope  objective  with  card- 
board or  paper  and  making  an  accurate  pointing  of  the 
horizontal  thread  upon  a  circumpolar  star  near  cul- 
mination. Having  made  a  satisfactory  pointing,  quickly 
shift  the  card  so  as  to  cover  the  lower  half  of  the  objective 
and  leave  free  the  upper  part,  when,  if  the  threads  are 
not  properly  adjusted  with  respect  to  the  objective, 
there  will  be  a  slight  vertical  displacement  of  the  star 


170  FIELD  ASTRONOMY. 

with  respect  to  the  thread,  and  this  must  be  corrected 
by  further  adjustment. 

Vertically  of  Threads. — To  make  the  threads  perpen- 
dicular to  the  rotation  axis,  point  the  telescope  at  a  ter- 
restrial mark,  and  turning  the  telescope  in  altitude 
with  the  slow-motion  screw,  note  whether  the  mark  in 
its  apparent  motion  up  and  down  the  field  of  view  runs 
exactly  along  the  thread.  Any  outstanding  error  in 
this  adjustment  may  be  removed  by  slightly  rotating 
in  its  own  plane  the  collar  which  carries  the  threads ;  but 
a  small  error  here  may  be  rendered  harmless  by  always 
pointing  the  telescope,  at  the  times  of  observation,  so 
that  the  stars  cross  the  same  part  of  the  field,  e.g.,  be- 
tween the  parallel  horizontal  threads. 

The  principal  errors  of  adjustment  that  remain  to 
be  considered  in  connection  with  the  use  of  a  transit 
instrument  are  three  in  number,  viz. :  The  azimuth  error, 
a,  is  the  angular  amount  by  which  the  rotation  axis 
deviates  to  the  south  of  due  west.  The  level  error,  b,  is 
the  angle  of  elevation  of  the  rotation  axis  above  the 
western  horizon.  The  collimation  error,  c,  is  the  amount 
by  which  the  angle  between  the  line  of  sight  and  the 
west  half  of  the  rotation  axis  exceeds  90°.  The  line  of 
sight  as  here  used  means  the  imaginary  line  passed 
through  the  Optical  centre  of  the  objective  and  the  mid- 
dle transit  thread,  or  through  the  mean  of  a  group  of 
transit  threads. 

75.  Theory  of  the  Instrument. — To  determine  the  rela- 
tion of  these  several  instrumental  errors  to  the  time,  T, 
at  which  a  star  will  pass  behind  a  given  transit  thread 


THE   TRANSIT  INSTRUMENT. 


171 


we  have  recourse  to  Fig.  13,  which  represents  a  projec- 
tion of  the  celestial  sphere  upon  the  plane  of  the  horizon. 
Z  is  the  projection  of  the  zenith,  P  of  the  pole,  H  of  the 
point  in  which  the  rotation  axis,  produced  toward  the 
west,  intersects  the  celestial  sphere,  and  5  is  the  projec- 
tion of  a  star  observed  at  the  moment  of  its  transit  over 


FIG.  13. — The  Transit  Instrument. 

a  thread  whose  angular  distance  from  H  is  measured 
by  the  arc  90°  +  c.  From  the  definitions  given  above,  c 
represents  the  collimation  of  the  particular  thread  in 
question,  and  similarly  b  and  a,  in  the  figure,  are  the 
level  and  azimuth  errors  above  defined.  The  symbol 
T  of  the  figure  represents  the  hour  angle  of  the  star, 
reckoned  toward  the  east,  d  is  the  star's  declination, 
90°  —  K  is  the  arc  HM,  and  A  is  the  distance  of  the  star 
from  the  meridian  measured  along  HS.  This  latter  arc 


172  FIELD  ASTRONOMY. 

must  not  be  confounded  with  the  diurnal  path  of  the 
star;  the  one  is  an  arc  of  a  great  circle  defined  by  the 
points  H  and  5,  while  the  other  is  a  small  circle  having 
its  pole  at  P.  Note  that  in  all  cases  the  symbols  here 
defined  represent  the  actual  magnitudes  of  the  arcs 
and  angles  on  the  sphere,  and  not  of  their  projections  on 
the  plane  of  the  horizon. 

From  the  spherical  triangle  PMS  we  obtain  the  rela- 
tion, 

cos  d  sin  T  =  sin  A  sin  $,  (143) 

and  from  the  triangle  ZHM  we  find, 

sin  K  =  sin  b  cos  £  +  cos  b  sin  £  sin  a.  (144) 

These  equations  may  be  greatly  simplified  by  substi- 
tuting arcs  in  place  of  sines  whenever  the  quantities  a 
and  b  are  so  small  that  their  cubes  and  higher  powers 
may  be  neglected,  and  we  shall  therefore  assume  that 
we  have  to  deal  with  an  approximately  adjusted  instru- 
ment, in  which  neither  of  these  quantities  much  exceeds 
10'.  On  this  supposition  the  point  H  is  so  nearly  the 
pole  of  the  meridian,  PZM,  that  we  may  put  sin  $  =  i 
and  "  =  <t>—d,  where  0  denotes  the  latitude  of  the  place 
of  observation,  and  our  equations  now  take  the  form, 

T  =  A  sec  d, 


Prom  the  figure  we  have  the  relation, 


THE  TRANSIT  INSTRUMENT.  173 

and  eliminating  A  and  K  between  these  equations  we  find, 
T  =  sin  (0  —  d)  sec  5.  a  +  cos  (0—5)  sec  d  .  b  +  sec  5.  c.  (146) 

Since  T  is  an  east  hour  angle,  we  have  also,  in  terms  of  the 
observed  time,  the  chronometer  correction,  and  the  star's 
right  ascension, 

r  +  J7  =  a-r,  (147) 

from  which  we  obtain,  by  the  elimination  of  T,  Mayer's 
equation  of  the  transit  instrument, 


a—  T  =  JT  +  sin  (0—5)  sec  d  .  a 

+  cos  (0-5)  sec  5  .  6  +  sec  5  .  c,     (148) 

or,  as  it  is  usually  written, 

a-T  =  4T  +  Aa  +  Bb  +  Cc,  (149) 

where  the  capital  letters  are  introduced  as  abbreviations 
for  the  coefficients  given  above,  i.e., 

A  =sin  (0-5)  sec  5,  B  =  cos  (0-5)  sec  5,  C  =  sec  5.  (150) 

Since  a,  T,  and  AT  are  expressed  in  time  (hours,  minutes, 
and  seconds),  it  is  customary  in  connection  with  this 
equation  to  express  a,  b,  and  c  in  seconds  of  time. 

76.  Discussion  of  Mayer's  Equation.  —  The  coefficients 
A,  B,  C  are  called  transit  factors,  and  when  many  obser- 
vations are  to  be  made  in  the  same  latitude,  0,  as  at  an 
observatory,  it  is  customary  to  tabulate  their  values 
with  the  declination  as  argument,  and  to  interpolate 
from  these  tables  the  values  of  the  factors  corresponding 
to  the  particular  stars  observed.  In  the  U.  S.  Coast  and 
Geodetic  Survey  Report  for  the  year  1880  there  may  be 


174  FIELD  ASTRONOMY. 

found  extensive  tables  of  this  kind  for  different  latitudes 
covering  the  whole  extent  of  the  United  States. 

In  the  use  of  such  tables  the  following  distinction 
must  be  carefully  observed:  Every  star  whose  distance 
from  the  pole  is  less  than  the  latitude  remains  continu- 
ously above  the  horizon  throughout  the  twenty-four 
hours,  and  during  this  period  crosses  the  meridian  twice, 
once  above  the  pole,  e.g.,  between  the  pole  and  zenith, 
and  once  below  the  pole,  e.g.,  between  the  pole  and  the 
northern  horizon.  The  latter  transit  is  usually  desig- 
nated sub  polo,  and  from  Fig.  13,  where  5'  represents 
the  star  5  near  its  transit  sub  polo,  it  may  be  seen  that 
its  coordinates  at  this  transit  will  be  obtained  by  sub- 
stituting in  place  of  the  a  and  90°  —  d,  corresponding 
to  5,  i2h  +  a  and  —  (90°—  d).  When  these  values  are 
introduced  into  Mayer's  equation  it  becomes,  for  stars 
observed  sub  polo, 

i2*  +  a-T  =  4T  +  A'a  +  B'b  +  C'c,  (151) 

where  the  new  transit  factors  have  the  following  values  : 

A'  =sin  (0+5)  sec  d,      Bf  =cos  (0+  d)  sec  d, 

C'  =  -sec5.  (I52) 

As  an  exercise  in  analysis  the  student  may  show  that 
the  transit  factors  for  a  star  above  and  below  the  pole  are 
connected  by  the  relations, 


A  +  A'  -2  sin  0,     £  +  £'  =  2cos0,     C  +  C'=o.     (153) 

Use  these  equations  to  derive  A'  ,  B',  C'  from  the  tabulated 
values  of  A,  B,  and  C. 

From  a  consideration  of  the  trigonometric  functions 


THE  TRANSIT  INSTRUMENT.  175 

that  enter  into  the  transit  factors  the  algebraic  signs  of 
these  factors  are  found  to  be  as  follows  for  places  in  the 
northern  hemisphere : 

Factor.  ABC 

South  of  Zenith +          +          +  - 

Zenith  to  Pole -          +          +  - 

Below  Pole +  -  + 

Note  that  in  every  case  the  transit  factors  for  a  given 
star  have  opposite  signs  above  and  below  the  pole,  and 
compare  with  this  statement  the  fact  that  stars  on  oppo- 
site sides  of  the  pole  move  in  opposite  directions,  east 
to  west  above  pole  and  west  to  east  below  pole. 

Query. — The  above  relations  of  sign  are  for  a  place 
in  north  latitude.  How  must  they  be  changed  to  adapt 
them  to  a  place  south  of  the  terrestrial  equator  ? 

In  explanation  of  the  double  set  of  signs  given  above 
for  C,  we  recall  what  was  shown  in  §  50,  that  a  reversal 
of  the  instrument  changes  the  sign  of  the  collimation 
constant,  c\  i.e.,  90°  —  c  is  substituted  for  the  90° +  £  of 
Fig.  13,  by  lifting  the  axis  out  of  the  wyes  and  replacing 
it,  turned  end  for  end.  It  is  customary  to  ignore  this 
change  of  sign  in  c,  and  to  represent  its  effect  in  Mayer's 
equation  by  changing  the  algebraic  sign  of  C  when  the 
instrument  is  reversed;  e.g., 

For  Circle  W C(  +  c)  =  (  +  Qc 

For  Circle  E C(-c)  =  (-C)c 

The  collimation  constant,  c,  may  be  either  positive  or 
negative,  depending  upon  the  adjustment  of  the  instru- 
ment ;  but  it  retains  the  same  sign  in  both  positions  of  the 


176  FIELD  ASTRONOMY. 

circle,  while  the  collimation  factor,  C,  is  positive  (above 
pole)  when  the  circle  end  of  the  axis  points  west,  negative 
when  it  points  east. 

77.  Choice  of  Stars.  —  In  the  right-hand  member  of 
Mayer's  equation,  as  printed  on  page  173,  there  are  in- 
volved four  unknown  quantities,  AT,  a,  b,  and  c,  one  of 
which,  6,  the  inclination  of  the  axis  to  the  plane  of  the 
horizon,  is  always  to  be  determined  by  some  mechanical 
method,  e.g.,  the  use  of  a  spirit-level.  The  collimation 
constant,  c,  may  also  be  determined  mechanically  (see 
§  84),  but  for  the  present  we  shall  assume  that  this  has 
not  been  done  and  that  the  instrumental  constants  a 
and  c,  as  well  as  the  clock  correction  AT,  are  to  be  deter- 
mined from  observations  of  stars.  Since  there  are  three 
quantities  to  be  thus  determined,  there  must  be  at  least 
three  observations,  and  it  is  practically  convenient  to 
make  four  the  minimum  number  instead  of  three;  ob- 
serving two  stars  Circle  E.  and  two  Circle  W.  for  the  sake 
of  a  good  determination  of  the  collimation,  c,  through 
the  reversal  of  the  instrument.  The  stars  thus  chosen 
should  not  all  lie  on  the  same  side  of  the  zenith,  but 
should  be  distributed  on  both  sides,  so  as  to  make  the 
sum  of  their  azimuth  factors  as  small  as  possible.  When 
I A  =o,  the  effect  of  the  azimuth  error,  a,  is  completely 
eliminated,  and  a  nearly  complete  elimination  may 
usually  be  obtained  by  care  in  the  selection  of  stars.  In 
the  example  of  §  78  this  condition  is  approximately 
satisfied  by  the  four  stars  marked  a',  b' ',  d' ',  e' ',  and  the 
student  after  tracing  through  the  reduction  there  given, 
should  note  that  if  the  azimuth  star,  i  H.  Draco.,  were 


THE  TRANSIT  INSTRUMENT.  177 

dropped  and  the  azimuth  error  entirely  ignored,  the 
resulting  value  of  AT  would  be  substantially  the  same 
as  is  obtained  when  the  azimuth  error  is  taken  into 
account.  In  this  case,  therefore,  an  accurate  deter- 
mination of  a  is  of  little  consequence. 

78.  Example. — Ordinary  Determination  of  Time. — The 
following  example,  taken  from  the  time  service  of  the 
Washburn  Observatory,  0  =  43°  4'  37",  illustrates  the 
record  and  reduction  of  a  set  of  transit  observations.  In 
addition  to  the  date  and  the  measured  inclination,  6, 
of  the  horizontal  axis,  given  in  the  column  of  Constants 
for  the  two  positions  of  the  instrument,  Circle  W.  and 
Circle  E.,  the  observed  data  are  contained  in  the  three 
columns  marked,  at  the  foot,  with  Roman  numerals, 
I,  II,  III.  The  observed  times  of  transit  given  in  III 
are  each  the  mean  of  the  observed  times  of  transit  of 
the  given  star  over  15  threads,  and  in  the  reduction  the 
collimation  constant,  c,  is  assumed  to  refer  to  the  mean 
of  these  threads  instead  of  to  the  middle  thread.  Note 
that  this  particular  convention  with  regard  to  c  can  be 
adopted  only  when  each  star  is  observed  over  precisely 
the  same  set  of  threads  as  every  other  star.  The  failure 
to  observe  a  single  star  at  its  transit  over  one  of  the 
threads  will  require  either  the  rejection  of  the  transits 
of  other  stars  observed  at  this  thread,  or  a  determination 
of  thread  intervals  and  a  ' ' reduction  to  the  mean  thread" 
for  which  reference  may  be  made  to  Appendix  7,  U.  S. 
Coast  and  Geodetic  Survey,  Annual  Report  for  1897-98. 

The  remaining  columns  are  marked  with  Arabic  nu- 
merals, showing  the  order  in  which  they  are  reached  in 


178  FIELD  ASTRONOMY. 

the  computation.  Of  these  columns  i  and  2  are  obtained 
from  the  almanac  (in  this  case  the  Berliner  Astronomisches 
Jahrbuch,  plus  the  corrections  given  in  Astronomische 
Nachrichten,  No.  3508).  The  declinations  are  taken  to 
the  nearest  minute  only,  while  the  right  ascensions  are 
accurately  interpolated  for  the  instant  of  the  star's  transit 
over  the  local  meridian,  i.e.,  0.3  day  after  their  transit 
over  the  meridian  for  which  the  almanac  is  constructed. 
The  third  star,  being  observed  sub  polo  (and  before  mid- 
night), was  observed  half  a  day  before  its  transit  over 
the  local  upper  meridian,  and  its  right  ascension  is  there- 
fore interpolated  for  an  instant  0.2  day  before  its  transit 
over  the  Berlin  meridian. 

The  transit  factors  contained  in  columns  4,  5,  and  6  were  inter- 
polated from  tables  of  such  factors,  and  the  products  contained  in 
columns  7  and  8  were  next  filled  in  by  the  use  of  Crelle's  multiplication 
tables.  It  may  be  noted  that  the  effect  of  diurnal  aberration  shown 
in  column  7  has  already  been  found  (§  27)  to  be  — os.o2i  cos  (f>  sec  5, 
which,  for  the  given  latitude,  is  equal  to— os.oi5  C,and  the  collimation 
factor  C  was  employed  in  computing  the  correction.  These  corrections 
were  next  added  mentally  to  the  numbers  contained  in  III,  and  the 
resulting  times  subtracted  from  the  right  ascensions  in  i,  thus  giving 
the  absolute  terms  of  the  equations  numbered  9.  The  first  members 
of  these  equations,  3,  4,  6,  are  obviously  derived  from  Mayer's  equa- 
tion. 

We  have  now  five  equations  involving  only  three  unknown  quan- 
tities and  presenting,  therefore,  a  case  for  the  application  of  the  Method 
of  Least  Squares.  A  rigorous  solution  by  that  method  furnishes  the 
following  values  of  the  quantities  sought: 

AT  =  +2m57s.oio,  a= +08.858,  c=+os.g66. 
But  such  a  solution  is  rather  laborious,  and  a  simple  method  of  obtain- 
ing approximately  accurate  results  is  indicated  under  the  heading 
Solution,  where  the  symbols  at  the  left  indicate  the  manner  in  which 
the  successive  equations  are  derived.  Equation  k'  is  derived  from 
*'  by  dividing  through  by  the  coefficient  of  c,  and  /'  is  similarly  de- 
rived from  hr ,  using  the  coefficient  of  AT ',  as  divisor  and  substituting 
in  place  of  c  its  value  given  by  k' .  Equation  mf  is  obtained  from  c' 
by  substituting  in  place  of  AT  and  c  their  values  as  given  in  kf  and  /'. 


THE  TRANSIT  INSTRUMENT.  179 

The  value  of  a  furnished  by  this  equation  when  substituted  in  k'  and 
/'  gives  definitive  values  of  AT  and  c,  all  of  which  are  entered  in 
the  column  of  Constants. 

By  means  of  these  values  of  a  and  c,  columns  1 2  and  1 3  are  filled 
up  and  the  sum  of  the  corrections  contained  in  columns  7,  8,  12,  13, 
is  entered  in  14  and  added  to  the  corresponding  numbers  in  III,  thus 
furnishing  the  corrected  times  contained  in  15.  Only  the  seconds 
are  entered  here,  since  the  minutes  remain  unchanged.  The  indi- 
vidual values  of  the  clock  correction  contained  in  16  are  now  obtained 
by  subtracting  15  from  i,  and  their  agreement,  one  with  another,  is 
a  check  upon  the  accuracy  of  the  entire  work,  both  observations  and 
computations.  For  the  sake  of  this  check  it  is  better  to  proceed  as 
is  here  done  than  to  rely  upon  the  value  of  AT  furnished  by  the  solu- 
tion of  the  equations.  The  numerical  work  here  shown  is  greatly 
facilitated  by  the  use  of  a  slide-rule  or  an  extended  multiplication 
table  such  as  that  of  Crelle. 

It  may  readily  be  seen  from  the  course  of  the  above 
solution  that  the  collimation,  c,  is  obtained  from  the 
four  observations  marked  a',  6',  df,  e' ,  while  the  azimuth, 
a,  is  furnished  by  the  third  observation.  A  star  near 
the  pole,  like  i  H.  Draco.,  is  introduced  into  the  observ- 
ing programme  solely  to.  determine  a,  and  with  refer- 
ence to  this  use  it  is  called  an  azimuth  star,  while  the 
others  are  known  as  clock  stars,  since  it  is  they  that  deter- 
mine the  value  of  AT.  As  there  is  always  a  possibility 
of  disturbing  the  azimuth,  i.e.,  changing  a  in  the  act  of 
reversing  the  instrument,  there  should,  in  all  strictness, 
be  two  values  of  a  determined,  one  for  Circle  W.  as  well 
as  the  one  above  found  from  the  observation  of  a  star 
Circle  E. ;  but  in  the  present  case  it  may  readily  be  seen 
that  there  was  no  such  disturbance,  since  the  value  of  a 
for  Circle  E.  brings  into  perfect  agreement  the  values 
of  AT  furnished  by  the  two  stars  observed  Circle  W., 
although  their  azimuth  factors  are  widely  different. 

Whenever  necessary,  introduce  into  the  solution  two 
azimuths,  one  for  each  position  of  the  instrument,  as 


180 


FIELD  ASTRONOMY. 


Constants. 

<N     Tf   10\O     HI 
.O     M  00     ON  O 

O    O    O    O    t— 

rO   »-O      Q      ^       M 

" 

(M     M     (N    IOOO 
^    0     0     0     O     ON 

10  10  10  10  10  O 

M 

» 

^  cooO    HI    -^-  O 
M    CO  O\  N     N 

g     O    NO      M     10   <M 

•a 

00    10  CO  ON  ON 

<N)    00      H4      CO    O 

te>i 

0 

U 

Tj-  \O     <N    IO  VO     HI 
co         to  co  '^- 

10  ON  COOO    O 
Tf  co  CONO    -<fr 

S 

3 

in 

+  +  +     1      1 

d 

CO 

«"               CMO      M      M        M 

? 

M    Tj-  ON  M    ON 
•    o    o    *O  CN    M 
w      00 

1       1     +     1       1 

co  O    O  O  NO 
10\O    ON  O    O 

+    1    +    1    + 

•—  JQ 

<M     CO  O     <N1     <N 

^  0    0    H,    o    0 

Q< 

1  1  +  1  1 

•= 

CONO    O    f^  ON 

CO         co  co  ^  HH 

— 

g  t^  cooo   CN   ON  HH 

£« 

•W3 

^^HWf£)S 

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THE  TRANSIT  INSTRUMENT.  181 

unknown  quantities.     It  is  not  necessary  to  introduce 
two  collimations. 

79.  Methods  of  Observation. — A  clock  or  chronometer 
is  an  indispensable  auxiliary  to  a  transit  instrument,  and 
an  observation  with  the  latter  consists  in  determining, 
as  accurately  as  may  be,  the  chronometer  time  at  which 
a  particular  star  transits  over  one  or  more  of  the  threads. 
In  the  best  astronomical  practice  a  recording  machine, 
called  a  chronograph,  is  used  in  this  connection,  but  we 
shall  here  suppose  the  observer  not  to  be  provided  with 
a  chronograph  and  constrained,  therefore,  to  use  the 
older  method  of  observing  by  eye  and  ear.  In  this 
method  the  observer  picks  up  the  beat  of  the  chronometer, 
i.e.,  counts  mentally  the  tick  corresponding  to  each  suc- 
cessive second,  i,  2,  3,  4,  etc.,  and  while  thus  counting 
looks  into  the  telescope  and  watches  the  progress  of  the 
star  across  the  field  of  view,  noting  its  position  at  the 
instant  of  each  counted  beat.  If,  by  any  chance,  the 
star  should  appear  exactly  behind  a  thread  at  the  instant 
when  the  counted  beat  was  26,  the  time  of  transit  over 
this  thread  would  be  recorded  26.0  seconds,  and  the  cor- 
responding hour  and  minute  subsequently  determined 
by  looking  at  the  face  of  the  chronom- 
eter. It  will  usually  happen,  however, 
that  the  star  passes  behind  the  thread 
between  two  chronometer  beats  instead 
of  simultaneously  with  one  of  them, 
somewhat  as  shown  in  Fig.  14,  where  FlG>  i4._Transits 
there  is  indicated  the  position  of  the  star  by  Eye  and  Ear' 
with  respect  to  the  thread  at  26"  and  at  27s,  as  noted 


182  FIELD  ASTRONOMY. 

and  temporarily  remembered  by  the  observer.  From 
the  manner  in  which  the  thread  divides  the  space  between 
the  two  star  images  it  is  evident  that  the  actual  transit 
over  the  thread  occurred  at  26.4s,  and  it  should  be  so 
recorded.  The  fraction  of  a  second  depends  upon  the 
observer's  estimation  (an  estimate  of  space  seen  in  the 
telescope  and  not  time  as  counted  by  the  ear),  and  a 
skilled  observer  should  be  able  to  follow  a  star  in  its 
progress  across  the  field  of  view,  observing  and  recording 
to  the  nearest  tenth  of  a  second  the  times  of  transit  over 
as  many  threads  as  may  be  desired,  without  taking  the 
eye  from  the  telescope  during  the  process.  He  should, 
while  watching  the  star,  give  no  heed  to  the  hour  and 
minute,  but  concentrate  attention  upon  the  seconds 
and  fractions  of  a  second,  until  the  transit  over  the  last 
thread  has  been  recorded;  then,  still  counting  seconds, 
let  him  look  back  at  the  face  of  the  chronometer  and 
note  if  the  time  there  shown  by  the  seconds  hand  agrees 
with  his  count.  This  is  called  checking  the  beat,  and 
if  it  checks  properly,  the  minute  and  hour  corresponding 
to  the  last  observation  should  be  written  down  as  a  part 
of  the  record. 

80.  Precision  of  the  Results.  —  By  the  method  above 
outlined  a  skilled  observer  may,  from  the  mean  of  several 
threads,  determine  the  time  of  a  star's  transit  within  very 
small  limits  of  error;  e.g.,  there  is  found  for  the  probable 
error  of  a  transit  of  a  single  star  over  the  mean  of  from 
10  to  15  threads,  some  os.o2  or  os.o3.  But  this  apparent 
precision  is  in  some  degree  fallacious,  for  most  observers 
possess  individual  peculiarities,  called  personal  equation, 


THE   TRANSIT  INSTRUMENT.  183 

by  which  they  tend  to  observe  all  stars  either  too  soon 
or  too  late,  by  a  nearly  constant  amount,  and  the  proba- 
ble error  of  a  transit  based  upon  the  agreement  of  indi- 
vidual results,  one  with  another,  furnishes  no  indication 
of  the  presence  or  magnitude  of  this  constant  personal 
error. 

Closely  related  to  the  precision  attainable  in  esti- 
mating the  times  of  transit  of  a  star  over  the  threads  of 
an  instrument,  is  the  degree  of  accordance  to  be  expected 
among  the  values  of  AT  furnished  by  the  several  stars 
composing  a  set,  such  as  that  of  the  illustrative  example 
of  §  78.  The  range  of  values  there  exhibited,  while 
smaller  than  is  to  be  expected  from  a  beginner,  may  be 
regarded  as  fairly  typical  of  the  results  to  be  obtained  by 
an  experienced  observer  provided-  with .  a  good  instru- 
ment. See  in  this  connection  the  example  of  §  82,  where 
the  results  show  an  even  closer  but  by  no  means  abnor- 
mal agreement. 

81.  Personal  Equation.  —  The  personal  equation,  al- 
though a  real  and  oftentimes  a  considerable  source  of 
error,  is,  however,  of  small  consequence  save  where  the 
observations  of  different  persons  are  to  be  combined,  one 
with  another,  as  in  a  determination  of  longitude.  In  such 
cases,  however,  the  problem  of  personal  equation  must 
be  met  and  seriously  dealt  with,  and  various  devices  have 
been  employed  for  this  purpose;  e.g. :  (i)  An  exchange  of 
observers  at  the  middle  of  the  work  in  question,  so  that 
its  first  half  may  be  affected  with  the  personal  error  in 
one  direction  and  the  second  half  in  the  opposite  direc- 
tion, thus  eliminating  this  influence  from  the  mean. 


184  FIELD  ASTRONOMY. 

(2)  The  determination  of  the  exact  amount  of  the  per- 
sonal equation  for  each  observer,  by  means  of  so-called 
personal-equation  machines,  is  sometimes  attempted; 
but  at  present  the  best  device  for  the  elimination  of 
personal  equation  seems  to  be :  (3)  The  Repsold  Transit 
Micrometer,  an  apparatus  in  whose  use  the  methods  of 
observing  above  set  forth,  §  79,  are  completely  aban- 
doned, and  as  a  substitute  for  them  the  observer,  while 
looking  into  the  telescope,  seeks  to  keep  the  image  of  a 
star,  as  it  moves  across  the  field,  constantly  covered  by 
a  micrometer  thread,  which  he  manipulates  with  his  fin- 
gers and  which  is  so  connected  with  a  chronograph  as 
to  give  an  automatic  record  of  the  star  transits.  The 
experience  of  the  Prussian  Geodetic  Institute  indicates 
that  in  this  mode  of  observing,  personal  differences 
between  observers  are  nearly  annihilated. 

82.  Reversal  of  the  Instrument  upon  Each  Star. — A 
method  of  using  a  transit  instrument  introduced  into 
general  practice  in  connection  with  the  transit  microm- 
eter, but  which  may  be  equally  well  applied  with  the 
ordinary  chronographic  or  eye-and-ear  methods,  con- 
sists in  noting  the  time  of  transit  of  a  star  over  a  group 
of  threads  placed  at  some  little  distance  from  the  centre 
of  the  field,  then,  after  quickly  reversing  the  instrument, 
to  observe  the  same  star  again  on  the  same  threads  in 
their  new  position.  It  is  obvious  that  the  effect  of 
collimation  is  thus  completely  eliminated  from  the  mean 
of  the  observations  on  each  thread  and  therefore  from 
the  general  mean  of  the  observed  times.  This  elimina- 
tion, while  an  important  advantage  of  this  mode  of  ob- 


THE  TRANSIT  INSTRUMENT.  185 

serving,  is  far  from  being  the  only  one,  and  a  considerable 
number  of  sources  of  error  which  have  not  been  con- 
sidered above,  but  which  are  dealt  with  at  length  in  the 
larger  treatises,  such  as  Chauvenet,  Spherical  and  Practi- 
cal Astronomy,  are  equally  eliminated  by  the  reversal; 
e.g.,  inequality  of  pivots,  flexure,  thread  intervals,  and 
the  disturbance  of  the  spirit-level  incident  to  reversing 
it  upon  the  axis.  When  the  telescope  is  reversed  upon 
every  star  a  hanging  level  may  be  allowed  to  remain 
upon  the  axis  without  ever  being  reversed,  since  the 
level  readings  in  the  two  positions  of  the  axis  then  give 
its  mean  inclination,  which  is  the  datum  required  for  the 
reduction  of  the  star  transits. 

Whenever  it  can  be  employed  the  method  of  reversal 
upon  every  star  is  to  be  preferred  to  the  older  method 
illustrated  in  the  preceding  example,  but  it  requires 
special  facilities  for  quick  reversal  of  the  instrument 
without  disturbing  its  azimuth,  and  these  are  not  always 
present. 

The  following  is  an  example  of  the  record  and  reduc- 
tion of  such  a  series  of  observations,  made  with  the  same 
instrument  and  arranged  in  nearly  the  same  manner  as 
the  example  on  p.  180.  Each  star  was  observed  on  five 
threads  in  each  position  of  the  instrument,  and  a  value 
of  the  level  constant,  6,  was  determined  for  each  star 
from  readings  of  the  hanging  level,  taken  immediately 
before  or  after  the  observed  transits  in  each  position 
of  the  circle,  the  level  remaining  unre versed  upon  the 
axis  during  the  entire  set  of  observations. 


186 


FIELD  ASTRONOMY. 


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THE   TRANSIT  INSTRUMENT.  187 

83.  Determination  of  Azimuth  with  a  Transit  Instru- 
ment.— Let  the  line  of  sight  of  a  transit  instrument  be 
supposed  directed  accurately  upon  some  terrestrial  mark, 
and  the  telescope  then  turned  up  to  the  sky  and  the 
time  of  a  star's  transit  over  the  line  of  sight  observed. 
From  this  observed  time  the  hour  angle  of  the  star  may 
be  derived,  and  this  hour  angle,  in  connection  with  the 
known  declination  and  latitude,  will  determine  the  star's 
azimuth  at  the  instant  of  observation.  If  there  are  no 
instrumental  errors  present,  this  computed  azimuth 
will  be  the  true  azimuth  of  the  terrestrial  mark  at  which 
the  line  of  sight  was  originally  directed. 

This  simple  method  of  determining  azimuth  requires 
some  modifications  on  account  of  instrumental  errors,  but 
when  these  are  duly  taken  into  account  and  a  proper 
selection  of  stars  and  mark  is  made,  the  method  ranks 
as  the  best  of  all  known  ones  for  azimuth  determination. 
The  star  to  be  observed  should  be  very  near  the  pole, 
usually  Polaris,  and  if  the  chronometer  correction,  AT, 
is  accurately  known,  the  observation  may  be  made  at 
any  convenient  time,  e.g.,  the  time  at  which  the  star 
stands  directly  above  a  mark  already  established.  If 
the  chronometer  correction  is  not  well  determined,  the 
observation  should  be  made  when  the  star  is  near  elonga- 
tion, since  the  effect  upon  the  computed  azimuth  of 
an  error  in  the  assumed  AT  is  then  a  minimum.  But 
this  latter  procedure  requires  the  establishment  of  a 
special  mark  whose  azimuth  shall  be  very  approximately 
equal  to  that  of  the  star  when  at  elongation,  and  it  will 
often  be  more  convenient  to  determine  the  time  with 


188  FIELD  ASTRONOMY. 

the  required  accuracy,  e.g.,  one  tenth  of  a  second,  and 
thus  obtain  more  freedom,  in  the  choice  of  a  mark. 

A  transit  instrument  of  the  better  class  is  usually 
provided  with  an  eyepiece  micrometer,  i.e.,  one  or 
more  threads  parallel  to  the  fixed  transit  threads,  but 
capable  of  being  moved  to  and  fro  in  the  field  of  view 
by  a  screw  whose  axis  is  parallel  to  the  rotation  axis 
of  the  instrument.  This  screw  is  provided  with  a  grad- 
uated head  whose  readings  indicate  the  successive  posi- 
tions of  the  thread  and  measure  the  amount  of  its  motion 
between  consecutive  pointings  upon  the  star  and  mark. 
When  such  a  micrometer  is  present,  transits  of  the  star 
may  be  observed  over  its  threads  as  long  as  the  star 
remains  within  the  field  of  view,  and  many  comparisons 
between  star  and  mark  may  be  substituted  for  the  single 
one  above  supposed.  The  instrument  should  be  re- 
versed at  least  once  during  these  observations,  and  the 
inclination  of  its  axis,  b,  must  be  carefully  determined 
since,  as  will  appear  later,  the  level  error  has  an  important 
effect  upon  the  azimuth. 

84.  Theory  of  the  Method.  —  To  derive  from  the 
micrometer  readings  upon  star  and  mark  the  difference 
of  their  respective  azimuths  we  have  recourse  to  Fig.  15, 
which  represents  a  projection  of  the  celestial  sphere  upon 
the  plane  of  the  horizon.  5  and  M  represent  respectively 
the  star  and  the  mark,  Z  is  the  zenith,  and  the  spherical 
angle  SZM  is  the  required  difference  of  azimuth.  Let 
H  be  the  point  in  which  the  rotation  axis  of  the  instru- 
ment, produced  toward  the  west,  meets  the  celestial 
sphere,  and  the  arcs  90°  -  b,  90°  -f  c,  will  then  have  the 


THE  TRANSIT  INSTRUMENT. 


189 


same  significance  as  in  Fig.   13.     The  spherical  angles 
HZS  and  HZM  are  represented  by  the  symbols  90 


FIG.  15. — Azimuth  with  a  Transit  Instrument. 


and  90°  -f  w',  and  the  zenith  distance  of  the  star,  ZS,  by  z. 
From  the  triangle  H ZS  we  obtain, 

cos  (90°  +  c)  =cos  (90° -b)  cos  z 

+  sin  (90°  —  b)  sin  z  cos  (90°  +  w),     (154) 

which,  when  b  and  c  do  not  much  exceed  10',  is  equivalent 
to, 

c  +  b  cos  z  =  w  sin  z.  (J55) 

In  this  equation  we  substitute  90°  — h  in  place  of  z  and 
put  sec  h  =  i  -f  0",  and  it  becomes, 

w  =  c  +  c<r  +  b  tan  h.  (156) 

From  the  triangle  ZHM  we  find  in  a  similar  manner  for 
the  mark, 

(is?) 


190  FIELD  ASTRONOMY. 

When  the  mark,  M,  is  in  or  very  near  the  horizon,  as  it 
should  be,  the  last  two  terms  of  Equation  157  vanish 
and  we  obtain  by  subtracting  it  from  Equation  156, 


-A-A'  =w-w'  =c-c'  +  co-  +  b  tan  h.         (158) 

Let  k  represent  the  angular  equivalent  (value)  of  one 
re  volution  vof  the  micrometer  screw,  R  the  reading  of  the 
screw-head  corresponding  to  any  position  of  the  movable 
thread,  and  R0  the  particular  reading  at  'which  the  angle 
between  the  rotation  axis  of  the  instrument  and  the  line 
of  sight  defined  by  the  thread  equals  90°,  i.e.,  RQ  is  the 
reading  corresponding  to  c  =  o.  For  any  other  position 
of  the  threads  corresponding  to  the  reading  R  we  shall 
have 

c=±k(R-R0),  (159) 

where  the  ambiguous  sign  depends  upon  the  position  of 
the  instrument,  whether  Circle  W.  or  Circle  E.  For  any 
given  instrument  it  is  well  to  determine,  by  trial,  once 
for  all,  in  which  of  these  positions  the  readings  of  the 
micrometer  head  continuously  diminish  as  the  microm- 
eter thread  is  made  to  follow  the  diurnal  motion  of  a 
star  near  upper  culmination,  and,  with  reference  to  the 
sign  in  Equation  159,  designate  this  as  the  positive,  the 
other  as  the  negative,  position  of  the  instrument. 

Corresponding  to  the  positive  and  negative  positions, 
respectively,  let  Rl  and  R2  be  readings  of  the  micrometer 
head  when  the  micrometer  thread  is  pointed  upon  the 
same  fixed  object,  e.g.,  the  mark  whose  azimuth  is  to 


THE  TRANSIT  INSTRUMENT.  191 

be  determined;  we  shall  then  have  as  the  distance  of 
this  object  from  the  line  of  no  collimation, 

Positive  Position,       s=+k(Rl-RQ), 
Negative  Position,     s=-k(R2-R0), 
from  which  we  readily  obtain, 

2s=+k(R1-R2), 
2R0  =  Rl  +  R2. 

The  first  of  these  equations  determines  the  distance,  s, 
of  the  terrestrial  mark  from  the  collimation  axis  of  the 
instrument,  and  it  should  be  used  to  make  the  distance 
of  the  azimuth  mark  small,  by  properly  placing  the  instru- 
ment, whenever  an  azimuth  determination  is  to  be  made. 
The  second  equation  determines  R0,  and  through  R0  the 
collimation  corresponding  to  any  position  of  the  microm- 
eter thread  may  be  found ;  e.g.,  let  R  denote  the  reading 
of  the  micrometer  when  the  movable  thread  is  placed 
in  apparent  coincidence  with  any  fixed  thread  of  the 
transit  reticule,  then  will  the  collimation  of  this  thread 
be  given  by  Equation  159.  This  method  of  determining 
collimation  may  be  employed  in  connection  with  time 
determinations,  as  indicated  in  §  77. 

To  apply  these  equations  to  the  reduction  of  a  set 
of  azimuth  observations  we  let  5  represent  the  mean  of 
several  micrometer  readings  made  in  quick  succession 
upon  the  star,  and  similarly  we  represent  by  M  the  mean 
of  several  readings  to  the  mark.  Introducing  these 
quantities  into  Equation  159,  we  find  for  the  star  and 
mark,  respectively, 

RJ,         (162) 


192  FIELD  ASTRONOMY. 

and  substituting  these  values  in  Equation  158,  we  obtain 
A-A'  =  ±k\(S-M)+  o-  (S-RJl+btanh.  (163) 
This  equation  may  be  used  for  the  reduction  of  the  ob- 
servations; but  if  the  instrument  has  been  frequently 
reversed  during  the  progress  of  the  work,  it  will  be  more 
convenient  to  combine  in  one  computation  consecutive 
observations  in  its  positive  and  negative  positions. 

V 

Employing  the  subscripts  i  and  2  to  distinguish  obser- 
vations made  in  these  respective  positions,  we  obtain,  by 
taking  the  mean  of  the  resulting  equations,  Circle  W. 
and  Circle  E.,  and  introducing  a  correction  for  diurnal 
aberration, 


-  b  tan  h  +  Di.  Ab.     (164) 

In  this  equation  A  represents  the  mean  of  the  azi- 
muths of  the  star  at  the  several  times  of  observation,  and 
for  this  mean  there  may  usually  be  substituted  the  azimuth 
corresponding  to  the  mean  of  the  times  (see  §  68).  The 
level  constant,  6,  represents  the  mean  of  the  inclinations 
of  the  horizontal  axis  in  the  two  positions  of  the  instru- 
ment, and  it  should  be  noted  that  if  the  level  is  left  undis- 
turbed upon  the  axis  during  the  reversal,  the  resulting 
bubble  readings,  Circle  W.  and  Circle  E.,  will  give  this 
mean  inclination,  free  from  the  effect  of  inequality  of 
pivots.  In  the  case  of  a  hanging  level  all  necessity  for 
lifting  it  from  the  axis  or  in  any  way  disturbing  its  rela- 
tion to  the  instrument  is  thus  removed. 

85.  Diurnal  Aberration.  —  In  explanation  of  the  last 
term  of  Equation  164  we  note  that  the  precision  attain- 


THE  TRANSIT  INSTRUMENT.  193 

able  with  a  transit  instrument  is  sufficient  to  demand  a 
consideration  of  the  effect  of  diurnal  aberration,  and  the 
student  may  show  from  the  data  in  §  27  that  for  any 
star  near  the  pole  this  effect  is  fully  compensated  by 
adding  to  the  computed  azimuth  of  the  mark  the  cor- 
rection, 

Di.  Ab.  =  +  o".32  cos  0  sec  h.  (165) 

Since  0  and  h  are  very  nearly  equal  for  close  circumpolar 
stars,  this  correction  is  practically  constant  and  equal 
to  -f  o".32, 

86.  Example. — Azimuth  Determination  with  Transit. — 
The  following  example  represents  the  record  and  reduc- 
tion of  a  single  set  of  azimuth  observations  made  with 
the  large  transit  instrument  of  the  " broken"  type 
shown  in  the  Frontispiece.  Note  that  the  recorded 
sidereal  times  show  that  the  observations  were  made  at 
about  9  or  10  o'clock  A.M.,  2ih  or  22h  astronomical 
reckoning,  i.e.,  in  broad  daylight.  Values  of  the  instru- 
mental constants  and  other  data  required  for  the  reduc- 
tion follow  immediately  after  the  record,  the  value  of 
the  chronometer  correction,  AT,  having  been  determined 
for  this  purpose  from  time  observations  immediately 
following  the  azimuth  work.  At  the  time  of  the  azimuth 
observations  Polaris  was  near  upper  culmination,  and  an 
inspection  of  the  micrometer  readings  to  the  star,  shows 
that  they  diminish  progressively  for  Circle  W.,  which  is 
therefore  the  positive  position  of  the  instrument  and  is 
to  receive  the  subscript  i  in  the  reductions.  The  azimuth 
of  Polaris  is  computed  from  Equation  124. 


194 


FIELD  ASTRONOMY. 


POLARIS   AND    AZIMUTH    MARK. 

Wednesday,  May  7,  1902. 
Bamberg  Transit.     Chronometer,  S.     Observer,  C. 


Circle. 

Azimuth  Mark. 

Polaris. 

Remarks. 
Levels. 

Micrometer. 

Chronometer. 

Micrometer. 

Chronometer. 

E. 
W. 

I5'637 

.622 
.601 
.606 
.610 

15.083 
.067 
.061 
.064 
.087 

h.      m. 
I       8 

I       2O 

9.803 
10.019 
.189 

19.799 
•578 
•394 

h.   m.      s. 
I    12    47 
13    26 
13    56 

i    16   18 
16  59-5 
17  35 

Mark  very 
unsteady. 

W.       E. 
29.4    59.2 
60.4   30.8 

+  1-3 
h=44°  1  8' 

Instrumental  Constants. 
= +318.0  £  =  57". 570  d=o".5o 

REDUCTION. 


M2 

*5-6*5 

$ 

43°    4'  37"-° 

S2 

10  .004 

d 

88    47        1.7 

», 

15-°7 

a 

h.    m.     s. 
1    23       5.4 

T  +  AT 

I    15    41.2 

log(S1-S1) 

0.9816** 

t 

23   52   35-8 

log<7 

9-5990 

t 

358     8  57.0 

+  (52-M2) 

—  5.611 

cos  t 

9-999774 

-(s.-JiQ 

+  a(S2-Sj 

-4.518 
-3.807 

tan  <£ 
cot  d 

9.970825 
8.326946 

Sum 

-I3-936 

sec  <£ 

0.136417 

sin  / 

8  .  509169** 

log  Sum 

i  .  14414** 

p  \  B\ 

o  .008532 

log  \h 

I-459I7 

\  B2 

171 

. 

—tan  A 

6  .  981235** 

cos  <£ 

9.863 

sec  /i 

0.147 

logx 

8-297545 

0.32 

9-505 

log*2 

6-5951 

Di.  Ab.               9-5r5 

A 
Micrometer 

180°    3'i7"-54 
—  6   41    .15 

Level 

—  o      o  .65 

Di.  Aberration 

+  o     o   .33 

Azimuth 

179  56  36  .07 

THE   TRANSIT  INSTRUMENT.  395 

To  eliminate  any  error  that  might  exist  in  the  assumed 
value  of  a  revolution  of  the  micrometer  screw,  a  second 
set  of  comparisons  of  the  star  and  mark  was  made  a  half- 
hour  later  than  those  reduced  above,  when  the  star  was 
on  the  opposite  side  of  the  mark  and  at  an  approximately 
equal  distance  from  it.  The  resulting  value  of  the  azi- 
muth of  the  mark  was  A'  =  179°  56'  36". 20. 

When  the  highest  accuracy  is  required  a  considerable 
number  of  such  sets  of  observations  should  be  made, 
extending  over  at  least  three  or  four  days  and,  when  pos- 
sible, so  timed  that  the  star  will  be  observed  at  opposite 
points  of  its  diurnal  path,  i.e.,  near  its  upper  and  lower 
•culmination,  in  order  to  eliminate  errors  in  its  assumed 
right  ascension  and  declination.  A  study  of  the  errors 
of  the  micrometer  screw  should  also  be  made  (see  §  72), 
and  the  resulting  corrections  for  periodic  and  progressive 
error  applied  to  the  several  readings.  The  azimuth  of 
the  star  should,  in  general,  be  computed  with  six-place 
logarithmic  tables,  but  when,  as  in  this  case,  the  star  is 
very  near  the  meridian  five  places  of  decimals  are  quite 
sufficient. 

Query, — Is  it  legitimate  in  this  case  to  neglect  the 
corrections,  AA0,  represented  by  Equation  127? 

For  an  extended  treatise  showing  the  methods  used 
in  the  U.  S.  Coast  and  Geodetic  Survey  for  the  deter- 
mination of  time  and  azimuth  with  a  transit  instrument, 
reference  may  be  made  to  Appendix  7,  Annual  Report  of 
the  Survey  for  1897-98. 


REFERENCE   WORKS. 

FOR  a  more  detailed  treatment  of  the  problems  of 
spherical  and  practical  astronomy  than  is  contained  in 
the  preceding  pages,  the  advanced  student  may  consult 
with  profit  the  following  works : 

1.  Chauvenet.     A  Manual  of  Spherical  and  Practical  Astronomy. 
2  vols.     Philadelphia.     Various  editions. 

2.  Hayford.     Determination  of  Time,   Longitude,   Latitude,  and 
Azimuth.      Appendix  No.  7,  U.  S.  Coast  and  Geodetic  Survey.     Sixty- 
seventh  Annual  Report.     Washington.     1899. 

3.  Albrecht.     Formeln  und  Hulfstafeln  fur  Geographische  Orts- 
bestimmungen.     Leipzig.     Third  Edition.      1894. 

4.  Albrecht.     Anleitung  zum   Gebrauche  des  Zenitteleskops  auf 
den  Internationalen  Breitenstationen.     Berlin.     1902. 

5.  Bamberg.     Anweisung  zur  Behandlung  der  Universal  Instru- 
mente  und  Theodoliten  mit  mikroskopischer  Ablesung,  etc.     Berlin. 
1883. 

Of  the  above  works  No.  i  is  the  standard  treatise  upon  the  subject; 
an  elaborate  manual  known  and  used  among  astronomers  of  every 
land.  No.  2  is  much  more  limited  in  its  scope,  but  presents  well  the 
methods  in  use  in  the  U.  S.  Coast  and  Geodetic  Survey.  No.  3  pre- 
sents similarly  the  current  German  practice  and  is  accompanied  by  a 
valuable  series  of  numerical  tables.  No.  4  is  a  special  monograph, 
and  No.  5  a  trade  pamphlet  presenting  details  of  the  use  and  care  of 
geodetic  instruments  not  readily  accessible  elsewhere. 

196 


TABLES 


INTRODUCTION  TO  THE  TABLES. 

§  87.  The  following  tables  are  intended  for  use  in 
connection  with  rough  and  approximate  determinations 
of  azimuth,  latitude  and  time.  They  are  sufficiently 
accurate  for  all  such  purposes,  the  quantities  inter- 
polated from  each  table  being  in  general  reliable  to 
within  one  unit  of  the  last  decimal  place  there  given, 
as  is  shown  in  the  several  illustrative  examples.  Even 
this  degree  of  precision  may  be  considerably  enhanced 
by  heeding  the  sign  + ,  in  many  places  printed  after  a 
tabular  number  and  implying  that  the  number  thus 
marked  is  to  be  increased  half  a  unit,  e.g.,  for  9+  read 
9.5.  To  the  limit  of  precision  thus  indicated  they 
render  the  observer  practically  independent  of  an 
almanac  for  the  reduction  of  observations  of  stars. 

For  an  explanation  of  the  tabular  numbers  reference 
may  be  made  to  the  sections  of  the  text  indicated  above 
the  respective  tables  and  to  the  following  explanation 
of  quantities  not  adequately  treated  in  the  text. 

Table  2,  n.     See  below  under  Table  5. 

Table  2,  V.  In  addition  to  the  use  of  V  set  forth 
in  §  1 9  the  following  application  may  occasionally  prove 
convenient:  The  solar  ephemeris,  i.e.,  the  right  ascen- 
sion and  declination  of  the  sun,  the  equation  of  time, 
etc.,  nearly  repeats  itself  on  corresponding  dates  in 

199 


200  FIELD    ASTRONOMY. 

successive  years,  although,  on  account  of  the  varying 
relation  of  the  calendar  year  to  the  true  solar  year, 
this  correspondence  is  not  sufficiently  exact  to  permit 
one  almanac  to  be  used  directly  as  a  substitute  for 
another.  By  taking  account  of  this  varying  relation  of 
the  calendar  to  the  true  solar  year  we  may,  however, 
make  such  use  of  a  substitute  almanac,  and  for  that 
purpose  we  add  to  the  calendar  date  for  which  any 
given  quantity  is  to  be  interpolated,  the  longitude  cor- 
rection, X,  of  §  17,  and  a  further  time  correction  entirely 
similar  to  A,  but  obtained  from  the  formula, 

Time  Correction  =  VA  —  V  Y, 

where  VY  denotes  the  tabular  value  of  V  for  the  given 
year  and  VA  is  the  value  of  V  for  the  year  of  the 
almanac  that  is  to  be  used.  Interpolate  the  required 
quantities  with  the  corrected  time  thus  obtained.  For 
example,  let  it  be  required  to  obtain  the  sun's  declina- 
tion, the  equation  of  time  and  the  right  ascension  of  the 
mean  sun  on  May  i,  1910,  at  1:30  P.M.  Central  Standard 
Time,  using  for  this  purpose  an  almanac  for  the  year 
1905.  We  proceed  as  follows: 


i9°5>  VA 

82.691 

Time  corr.  +  A 

+   oh 

53m.3 

1910,  VY 

82.904 

Stand,  time 

i 

30    .0 

Greenwich  t. 

2 

23    -3 

VA-VY 

-r0.2I3<f,= 

d 

14° 

59'      5" 

Time  coir. 

—  5h    6m.  7 

E 

2      57  -32 

A 

+  6      o    .0 

Q 

2 

35     22  -16 

From  the  alma-n-ac  for  1910  we  find  for  the  given  time 
=  14°  59'  i2".8,      E=2m  57s.i6,      Q  =  2h  35m  22s.i7, 


TABLES.  201 

and  the  discrepancies  between  these  values  and  those 
found  above  fairly  illustrate  what  may  be  expected  in 
other  cases. 

Note  that  for  any  given  meridian  and  almanac  the 
Time  Correction  +L  +oh  5 3™. 3  in  the  example,  is  con- 
stant for  a  year,  and  that  when  its  value  has  once  been 
computed  and  written  in  the  margin  of  the  page,  in  all 
that  relates  to  the  solar  ephemeris  an  old  almanac  is  as 
convenient  for  use  as  the  current  one.  But  only  quan- 
tities pertaining  to  the  sun,  e.g.,  d,  a,  E,  Q,  etc.,  can  be 
correctly  found  by  the  method  given  above. 

Table  5.  Right  Ascensions  and  Declinations  of  the 
fixed  stars  are  best  obtained  from  the  almanac  of  the 
year  in  which  they  are  observed,  but  for  the  convenience 
of  the  observer  not  provided  with  such  an  almanac,  there 
are  here  printed  tables  from  which  may  be  derived  the 
coordinates  of  Polaris  and  fifty  southern  stars,  all  of 
which  are  bright  enough  to  be  readily  observed  in  the 
telescope  of  an  engineer's  transit. 

In  order  that  such  tables  shall  correctly  represent 
the  positions  of  the  stars  they  must  take  account  of 
three  classes  of  variation  to  which  the  coordinates  are 
subject,  viz: 

A.  The   Annual    Variation,     a   nearly    uniform    pro- 
gressive change  whose  amount  for  each  right  ascension 
and  declination  is  shown  under  the  heading,  Ann.  Var., 
in  Table  5  4. 

B.  An    oscillating  change,   aberration,    solar    nuta- 
tion, that  runs  through  its  entire  series  of  values  each 
year. 


202  FIELD    ASTRONOMY. 

C.  A  slower  oscillating  change,  lunar  nutation,  whose 
period  is  approximately  18.7  years. 

Table  5^  shows  for  each  star,  in  addition  to  its  name 
and  stellar  magnitude,  §  21,  its  right  ascension  and 
declination  on  that  date  of  the  year  1900  which  is 
printed  in  the  next  to  the  last  column  of  the  table. 
The  approximate  coordinates  on  the  same  date  in  any 
subsequent  year,  1 900  +  T,  may  be  found  by  adding  to 
the  tabular  ao,  #o,  T  times  the  corresponding  Ann.  Var. 
The  printed  a0,  #o,  include  the  effects  above,  designated 
A  and  B,  but  take  no  account  of  C,  and  in  order  to 
include  its  influence  we  must  have  recourse  to  Table  2 
where  the  column  n  gives  a  correction  to  the  time  inter- 
val, T,  sufficient  to  take  into  account  the  major  part  *of 
the  nutation  effect,  e.g.,  the  actual  a  and  d  are  given 
by  the  equations, 

a  =  a0+(T+tt)  Ann.  Var.       d  =  dQ+(T+n)  Ann.  Var. 

where  T  is  always  an  integral  number  of  years. 

The  declination  thus  computed  may  be  assumed  to 
remain  constant  throughout  the  year,  and  the  right 
ascension  may  similarly  be  assumed  constant  for  a 
month  preceding  and  following  the  date  named  in  the 
table.  For  remoter  dates  apply  to  the  computed  a  a 
correction  interpolated  from  Table  5B.  For  an  explana- 
tion of  Table  $c  and  the  column  M 0  to  which  it  relates, 
see  §  32. 

In  illustration  of  the  foregoing  we  shall  find  the  co- 
ordinates of  Antares,  No.  34,  for  May  i,  1911,  as  follows: 

T+n=  ii  —0.3  =  10.7  years. 


TABLES. 


203 


10. 7  Ann.  Var. 


Epoch  —  May  1  =  87  days. 

10.7  Ann.  Var. 


16 


+  39  -3 
-  o  .3 

23       58  -5 


-26°    1 2'.  5 


—26 


14. 


The  almanac  furnishes  as  the  coordinates  of  Antares 
on  this  date 

a=i6h  23m  58^.45  £=-26°  14'  1  6". 

Table  6.  The  right  ascension  of  Polaris  is  subject  to 
such  large  and  rapid  variations  that  the  coordinates  of 
this  star,  are  given  under  a  form  very  different  from 
that  of  Table  5  .  In  Table  6A  there  are  given  for  the 
beginning  of  each  year  from  1905  to  1930  the  quantities 
«i,  #1,  which  include  the  effect  of  the  variations  above 
designated  A  and  C,  and  in  Table  6B  are  given  at  uni- 
form intervals  of  20  days  throughout  the  years  1910  and 
1930,  quantities,  a2,  §2,  which  represent  the  effect  of  B. 
Interpolate  for  the  given  year  and  day  the  several  quan- 
tities «i,  di,  a2,  $2,  (double  interpolation  in  6B)  and 
find  the  coordinates  of  Polaris  through  the  relations  : 


To  facilitate  interpolation  in  6A  there  is  added  to  6B  a 
column,  T,  showing  the  fraction  of  a  year  that  has  elapsed 
at  each  date. 

For  example,  we  find  for  Greenwich  mean  noon,  Octo- 
ber 15,  1905,  r  =  o.79, 


2 
26 


o8 

19 


88°    47'    48".  4 

20     .6 

88      48        9  . 


The  almanac  gives  for  Polaris  on  this  date, 

a  =  ih  26m  i98.5  d  =  88°  48'  9".!. 


204  FIELD   ASTRONOMY. 

§  88.  Differential  Coefficients.  —  Some  of  the  ends 
served  by  the  formulas  printed  on  p.  207  may  be  exem- 
plified as  follows:  Referring  to  the  method  of  deter- 
mining latitude  represented  by  Eq.  35,  it  is  apparent 
that  if  the  altitude,  h,  be  measured  too  great  by  i'  the 
computed  latitude,  <j>,  will  be  found  i'  too  small.  In 
the  notation  of  the  differential  calculus  this  relation  is 
represented  by  the  equation 

d(j>/dh  =  -i, 

a  relation  which  in  this  simple  form  holds  only  when  the 
body  is  in  the  meridian.  In  Eqs.  36,  37,  38,  if  h  be 
measured  i'  too  great  the  computed  azimuth,  A,  and 
time,  T,  will  be  to  some  extent  vitiated,  but  the  amount 
of  error  thus  produced  is  not  immediately  evident.  The 
ratio  of  the  resulting  errors,  dT,  dA,  to  the  dh  that  pro- 
duced them  is,  however,  given  by  the  differential  coeffi- 
cients -TTj  -jTj  and  the  analytical  equivalents  of  these 

expressions  as  well  as  others  of  frequent  use,  are  given 
on  p.  207.  By  a  proper  use  of  these  expressions  a  cor- 
rection for  small  errors,  e.g.,  o'  to  5',  whose  presence  in 
the  data  is  detected  only  after  a  long  reduction  has  been 
made,  may  be  readily  computed  without  the  necessity 
for  a  second  reduction  of  the  observations;  thus,  in 
the  example  solved  in  §  30,  if  it  be  desired  to  sub- 
stitute in  place  of  the  latitude  there  employed  an  im- 
proved value  o'.4  greater,  we  may  find  the  resulting 
change  in  AM  to  be 

dA 
JA    =        +o'.     =  -o/.sec       cot  t=  -o'. 


TABLES.  205 

using  for  sec  <£  and  cot  /  the  numerical  values  furnished 
by  the  original  reduction. 

An  equally  important  use  of  these  differential  coeffi- 
cients is  that  they  furnish  criteria  by  which  to  distin- 
guish between  favorable  and  unfavorable  conditions 
for  observations  of  a  given  type.  Thus  if  it  be  desired 
to  determine  time  from  an  observed  altitude  of  a  star, 
the  conditions  should  be  so  chosen  that  any  small  error 
in  the  observed  altitude  shall  have  a  minimum  effect 
upon  the  concluded  T,  or  the  hour  angle,  t,  from 
which  T  is  determined.  We  find  from  the  first  of  the 
differential  coefficients  on  p.  207. 

dt__  i 

dh         cos  (f>  sin  A' 

which  for  any  given  value  of  the  latitude,  <f>,  has 
its  minimum  value  when  sin  A  =  i,  i.e.,  the  observation 
should  be  made  upon  a  star  near  the  prime  vertical. 
Let  the  student  show  from  the  fourth  of  the  differential 
coefficients  that  in  the  determination  last  considered 
a  small  error  in  the  adopted  latitude  will  also  have  its 
minimum  effect  upon  the  resulting  chronometer  cor- 
rection if  the  observation  is  made  upon  a  body  due  east 
or  due  west. 


DIFFERENTIAL  COEFFICIENTS.  §88. 

dh 

—  =  —cos  (f)  sin  A  ....................  (f>  and  d  constant 

—  =_cos^.  ......................... 

d<p 

dh 

—  =  —  cos  h  tan  q  ....................  6  and  d        " 

dA 

d6 

~-  =  —  cos  cj)  tan  A  ....................  d  and  h         " 

dt 

dd 

j-  =  +cos  <?  tan  q  .....................  <£  and  & 

dd 

and  h         * 


^ 

—  =—  sec  0  cot  /  .....................  d  and 

d(j) 

dA 

—  =  +cos  d  sec  /j  cos  ^  ................   <£  and 

=  sin  (j)  +cos  <£  tan  A  cos  A 
_?=.^cos  ^,  sec  h  cos  A  ...............  <£  and 

^2/ 
-T 


—cos  (j)  cos  £  sec  A  cos  A  cos  g.  .  .  .  <£  and  d        " 


s^,>s^.- 

COS  (f)  /COS  d>\2    . 

— -fsm  d  sin  A  —  { 7  )  sm  2 A  •  <b  and  d         " 

cos  h  \cos  h/ 

=  —cos  d  cos  0  sec2  A  {sin  h  sin  .4  cos  q  -j-  cos  ,4  sin  t 

307" 


208 


FIELD  ASTRONOMY. 


ALTITUDE   NUMBERS. 

TABLE  IA—  VALUES  OF/. 


§32. 


h 

0° 

10° 

20° 

30° 

40° 

50° 

60° 

k 

0° 

0.766 

0.778 

0.815 

0.885 

.000 

.192 

1-532  T00 

0° 

2 

0.767 

0.783 

0.826 

0.903 

.031 

-245 

*-632   16 

2 

4 

0,768 

0.790 

0.839 

0.924 

.065 

-3°3 

T-748  "g 

4 

6 

0.770 

o-797 

0.852 

0.947 

.103 

-370 

z'884   61 

6 

8 

0-773 

0.806 

0.868 

0.972 

-145 

-445 

2-045  \Q\ 

8 

10 

0.778 

0.815 

0.885 

I.OOO 

.192 

-S32 

2.240  I95 

10 

TABLE  Is.— REFRACTION,  ETC. 


§§  29, 32 


A' 

R 

R' 

10° 

5'-i 

5'-o 

20 

2  -5 

2  .4 

30 

i  .6 

i-5 

40 

i  .1 

I   .0 

50 

0.8 

o-7 

60 

o-5 

0-5 

70 

o-3 

0-3 

80 

0  .2 

O  .2 

9° 

O  .0 

0  .0 

TABLES. 


209 


YEAR  NUMBERS. 

TABLE  2. 


§§  19,  32,  87. 


Year. 

V 

n 

Y 

F 

d 

y 

m 

I9°5 

82.691 

—  o.i 

+  14 

i.  020 

1906 

82.935 

0.2 

12 

1.016 

1907 

83.178 

°-3 

II 

I.  01  2 

1908 

82.420 

o-3 

13 

1.007 

1909 

82.663 

o-3 

12 

I.OO2 

1910 

82.904 

o-3 

+  10 

0.997 

1911 

83.146 

o-3 

9 

0.992 

1912 

82.386 

0.2 

12 

0.987 

*9J3 

82.627 

—  O.I 

IO 

0.982 

1914 

82.868 

+  0.1 

9 

0.978 

I9i5 

83.110 

O.2 

+  7 

o-973 

1916 

82.350 

o-3 

9 

0.968 

1917 

82.592 

0.3 

8 

0.964 

1918 

82.834 

°-3 

6 

0.960 

1919 

83.076 

o-3 

5 

0.956 

1920 

82.319 

0-3 

+  7 

0.952 

1921 

82.562 

O.2 

6 

0.948 

1922 

82.806 

+  0.1 

4 

0.944 

1923 

83-050 

—  o.o 

3 

0.940 

1924 

82.293 

0.2 

5 

0.936 

I925 

82.537 

0-3 

+  4 

0.932 

1926 

82.779 

0-3 

3 

0.927 

1927 

83.022 

°-3 

i 

0.923 

1928 

82.264 

°-3 

4 

0.919 

1929 

82.506 

o-3 

3 

0.914 

1930 

82.747 

—  0.2 

+  i 

0.009 

V  is  referred  to  the  Greenwich  meridian. 


210 


FIELD   ASTRONOMY. 


DAY  NUMBERS. 

TABLE  3. 
For  Greenwich  Mean  Noon. 


§3* 


Day  of  the 

D 

JF 

Month. 

Year. 

Jan.     6* 

7 

I?h        23m 

—  O.OO2 

21* 

22 

18         22 

0.002 

Feb.     5* 

20* 

v:      37 

52 

19            22 
20            21 

O.002 
O.OOI 

PP  for  D. 

Mar.    7 

67 

21            21 

—  O.OOI 

HI 

22 

82 

22            2O 

o.ooo 

i 

3m-9 

Apr.    6 

97 

23            20 

+  O.OOI 

2 

7  -9 

o 

II      »O 

21 

112 

o        19 

O.OOI 

May    6 

127 

I            18 

0.002 

4 

15   .8 

21 

142 

2            17 

0.003 

5 

19    ,7 

June    5 

157 

3        16 

0.004 

6 

23   -7 

x 

27     *O 

20 

172 

4         15 

0.005 

/     *  w 

July    5 

I87 

5         14 

0.005 

31   -5 

20 

2O2 

6        13 

0.005 

9 

35  -5 

Aug.-    4 

217 

7        ii 

0.004 

10 

39  -4 

19 

232 

8        10 

0.004 

Sept.    3 

247 

9          9 

0.003 

18 

262 

10          8 

O.002 

Oct.     3 

277 

ii          7 

-f  o.ooo 

18 

292 

12              6 

—  O.OOI 

Nov.    2 

307 

13          5 

O.OO2 

17 

322 

14          4 

0.003 

Dec.     2 

337 

15          4 

0.004 

17 

352 

16          3 

0.005 

32 

367 

17               2 

—  0.006 

*  In  leap  year  subtract  one  day  from  the  given  date  in  January  and  Feb- 
fuary  before  interpolating  from  this  table. 


Year. 


Ff 


TABLES. 

POLARIS   ORIENTATION. 

TABLE  4. 


211 


32,  33- 


t 

0« 

Var.  per  lm 

60 

Var.  per  lm 

i 

h.  m. 

/ 

/ 

/ 

/ 

h.  m. 

o   o 

-  0  + 

0.40 

+  70+ 

O.OO 

24   o 

o  30 

12 

0-39 

70 

0.04 

23  3° 

I    0 

24 

0.38 

68 

0.08 

23   o 

i  3° 

35 

o-37 

65 

O.I2 

22  30 

2    0 

-46  + 

o.35 

+  6!  + 

0.15 

22    O 

2   30 

56 

0.32 

56 

0.18 

21   30 

3   o 

65 

0.29 

50 

O.2I 

21    O 

3  3° 

73 

0.25 

43 

0.24 

2O   30 

4   o 

-80  + 

O.2O 

+  35+ 

0.27 

2O    O 

4  3° 

85 

0.15 

27 

0.29 

19   30 

5   ° 

89 

O.IO 

18 

0.30 

19   o 

5  3° 

9i 

0.05 

9 

0.30 

18  30 

6   o 

-92  + 

o.oo 

±  o± 

0.30 

18   o 

6  30 

91 

0.05 

9 

0.30 

17  30 

7   o 

89 

O.IO 

18 

0.30 

17   o 

7  3° 

85 

0.15 

27 

0.29 

16  30 

.8   o 

-80  + 

O.20 

-35- 

0.27 

16   o 

8  30 

73 

O.25 

43 

0.24 

i5  3° 

9   ° 

65 

O.29 

50 

O.2I 

15    0 

9  3° 

56 

0.32 

56 

0.18 

14  3° 

10   o 

-46  + 

o-35 

-61- 

0.15 

14   o 

10  30 

35 

o-37 

65 

O.I2 

J3  3° 

II    O 

24 

0.38 

68 

0.08 

13   o 

II  30 

12 

o-39 

70 

O.04 

12  30 

12    0 

-  o  + 

0.40 

-70- 

O.OO 

12    O 

t=M+Y+D. 


212 


FIELD    ASTRONOMY, 


TIME   STARS. 

TABLE  5A. 


32,  87. 


No. 

Star. 

Mag. 

at  1900. 

Ann. 
Var. 

di  1900. 

Ann. 
Var. 

I      Date. 

Mo 

h.  m.     s. 

s, 

0         / 

/ 

. 

h.m. 

1 

j  Ceti  

•3 

o  14  23 

+  3.06 

—     9    22  + 

+  O.  33 

Nov.  24 

8  o 

2 

ft  Ceti  .    . 

O 
2 

o  38  17  + 

1    O 

3.01 

—  18  32 

1           OO 

O.  33 

30 

o 

•7 

0  Ceti  

4 

O       O  I 

i  19    5 

o 
3.00 

**  o 
—  8  .41  + 

JJ 
0.31 

o 
Dec.  10 

i 

o 
4 

a  Piscium  ... 

4 

i  56  56 

o 

3.10 

+    2  17 

O 

0.29 

20 

o 

5 

d  Ceti...... 

4 

2  34  25 

3-°7 

-  o    6 

0.26 

29 

I 

6 

a  Ceti  

•7 

2    C7      4 

+  3.13 

4-  3  42 

+  0.24 

Jan.     4 

8  o 

7 

e  Eridani  .  .. 

<J 

4 

O  i         " 

3  28  14 

1    O*O 

2.82 

1           O      *T** 

-  9  48 

O.2I 

ii 

3 

8 

f  Eridani  .  .. 

3 

3  53  22  + 

2.80 

-13  48 

0.17 

18 

i 

9 

y  Eridiani  .. 

4 

4  31  20 

3.00 

-  3  33  + 

O.I3 

27 

3 

10 

ft  Orionis  .  .  . 

0 

5     9  45 

2.88 

—  8  19 

O.O7 

Feb.     6 

2 

ii 

K  Orionis  .  .  . 

2 

5  43     2 

+  2.84 

-  9  42  + 

+  O.O2 

IS 

80 

12 

ft  Can.  Maj.  . 

2 

6  18  19 

2.64 

-17  54  + 

—  O.O2 

24 

O 

I? 

Sirius  

—  I 

6  40  45  + 

2.64 

—  16  7? 

0.08 

Mar.    i 

2 

O 

14 

f  Can.  Maj. 

4 

6  59  15 

2.71 

OJ 

-15  29  + 

0.08 

6 

O 

I 

15 

TJ  Can.  Maj. 

2 

7  20    9+ 

2-37 

-29    7 

O.II 

ii 

2 

16 

p  Argus  

3 

8    3  18 

+  2.55 

-24    i 

-0.17 

22 

8   2 

17 

1  2  Hydrae  

4 

8  41  40+ 

2.84 

-13  ii 

O.22 

Apr.     i 

I 

18 

a  Hydrae  

2 

9    22    42 

2-95 

-  8  14 

O.26 

ii 

3 

19 

39  Hydrae  

4 

9  46  41 

2.88 

-14  23 

0.28 

17 

3 

20 

A  Hydrae.... 

4 

10    5  44 

2.92 

-ii  52 

0.30 

22 

2 

21 

v  Hydrae  .... 

3 

10  44  43 

+  2.96 

-15  40+ 

-0.3I 

May     2 

8   2 

22 

d  Crateris.  .  . 

4 

II    14    22 

3.00 

-14  14+ 

0.32 

10 

0 

23 

ft  Virginis  .  .  . 

4 

ii  45  3i 

3-i3 

-H     2    20 

0.34 

18 

0 

24 

f  Corvi  

3 

12    IO   41  + 

3.081 

—  16  59+ 

O.3.3 

24 

I 

25 

f  Virginis  .  .  . 

o 
3 

12  36  37 

O 

3-°4 

-  o  54 

oo 

0.33 

3i 

o 

26 

6  Virginis... 

4 

13     4  48 

+  3.10 

-   5    o+ 

-0.32 

June    7 

8  o 

27 

Spica  .  . 

i 

13   10  ^7  + 

3.16 

—  10    38  + 

O.3I 

ii 

o 

/ 

28 

£  Virginis  .  .  . 

3 

o     y  o  / 
13   29  38 

O 

3.05 

O 

-05  + 

o 

0.31 

13 

I 

29 

n  Hydrae  

4 

14     o  42  + 

3-4i 

—  26    12 

0.29 

21 

I 

3° 

/(  Virginis.  .  . 

4 

14  37  49  + 

3-i6 

-  5  i3  + 

0.26 

3° 

2 

31 

ft  Librae  .... 

3 

15  ii  40 

-r-3-22 

-  9    i 

—  O.22 

July     9 

8  i 

32 

v.  Serpentis.  . 

4 

15  44  26  + 

3-13 

-  3     7  + 

0.18 

17 

2 

33 

ft  Scorpii  .  .  . 

3 

15  59  40 

3.48 

-19  32 

0.17 

21 

2 

34 

Antares  .  .. 

i 

16  23  19+ 

3.67 

—  26    12  + 

0.14 

27 

2 

35 

TJ  Ophiuchi 

3 

17     4  41 

3-44 

[-15    36 

0.08 

Aug.    7 

O 

TABLES. 


213 


TIME  STARS. 

TABLE   5A. 


32,  87. 


No. 

Star. 

Mag. 

at  1900. 

Ann. 
Var. 

dt  1900. 

Ann. 
Var. 

Date. 

M9 

h.   m.    s. 

S 

o      / 

/ 

h.m. 

36 

£  Serpentis  .  . 

4 

17  3i  54  + 

+  3-43 

-15    20 

—  0.04 

Aug.  13 

83 

37 

v  Ophiuchi  . 

4 

17  53  34 

3-3° 

-   9  45  + 

—  O.OI 

19 

i 

38 

•n  Serpentis  .  . 

3 

18  16  10  + 

3-io 

-   2  55  + 

o.oo 

25 

0 

39 

a  Sagittarii  .  . 

2 

18  49     7 

3-72 

—  26  25 

+0.07 

Sept.    2 

2 

40 

f)  Sagittarii  .  . 

4 

19  iS  55  + 

3,48 

-18     2 

+  O.II 

9 

I 

4i 

T)  Aquila;  .  .  . 

4 

19  47  25  + 

+  3.06 

+  o  45 

-f  o.  15 

17 

8  i 

42 

ft  Capricorni 

3 

20  15  26+ 

3-37 

-IS     5  + 

0.18 

24 

I 

43 

s  Aquarii  .  .. 

4 

20    42    19 

3-25 

-  9  5i  + 

O.  22 

Oct.     i 

0 

44 

v  Aquarii  .  .. 

4 

21       4    12 

3-27 

—  ir  46+ 

0.24 

6 

2 

45 

/?  Aquarii  .  .. 

3 

21     26    2O  + 

3-i6 

-60 

O.26 

12 

I 

46 

a  Aquarii  .  .  . 

3 

22      O   42 

+  3-o8 

—  o  48 

+  0.29 

21 

80 

47 

£  Aquarii  .  .  . 

4 

22  23  44 

3-09 

-   o  31  + 

0.30 

26 

3 

48 

8  Aquarii  .  .  . 

3 

22  49  23  + 

3-i9 

—  16  21 

0.32 

Nov.    2 

i 

49 

;•  Piscium  

4 

23    12       2 

3-1* 

+    2  44  + 

°-33 

8 

0 

5° 

ID  Aquarii  .  .  . 

4 

23  37  35  + 

3-11 

-15     5  + 

o-33 

14 

2 

TABLE  5B.        §  87. 
Correction  to  R.A. 


Date. 

Act 

1  50  days  before 

100     " 

50    " 
Epoch  

s. 
-i-9 
-o-5 
+  0.3 
o.o 

50  days  after. 

100      " 

-0.7 

—  1.2 

TABLE  5C. 
Correction  to  M 0. 


§32. 


Leap  Year. 

Longitude. 

I. 

II. 

III. 

Jan. 

Mch. 

Feb. 

Dec. 

m. 

m. 

m. 

m. 

m. 

I2h    E. 

8 

4 

5 

6 

7 

6     E. 

7 

3 

4 

S 

6 

Greenwich 

6 

2 

3 

4 

S 

6h  W. 

5 

I 

2 

3 

4 

12      W. 

4 

0 

I 

2 

3 

214 


FIELD    ASTRONOMY. 


POLARIS. 
TABLE  6A. 


§87. 


Year. 

«, 

'i 

h   m   s 

o     in 

1905 

i  23  45 

88  47  35 

1906 

24   4  ig 

J7 
47  52    jg 

1907 

24   22 

48  10 

1908 

24  39 

48  29 

1909 

,   17 
24  56 

48  48  + 

1910 

19 

2  2 

88  49   9    2° 

20 

1911 

25  37 

49  29  + 

25 

2  I 

1912 

26    2 
28 

49  50  + 

1913 

26   30 

50  ii  + 

31 

21 

1914 

27    I 

50   32 

t             35 

20 

1915 

I   27   36 

88  50  52  + 

1916 

20 

51   I2     ,8 

1917 
1918 

28   52 
39 
29   31   ,. 

i  r  -s 

39 

i? 

1919 

3°   I0   3, 

52    5     ,6 

1920 

i  3°  47  34 

88  52  21+  I6 

1921 

31   21 

S2  37+  16 

1922 

31   52   2g 

S2  53+  I6 

1923 

32   20   ^ 

53   9+  I7 

1924 

32  44  23 

53  26+  i? 

1926 

1  33   7  „ 

33  28  20 

88  53  43+  Ip 
54   2    19 

1927 

33  48  J2 

54  2I    20 

1928 

34  10  2s 

54  40+  2o 

1929 

34  35  27 

55   J    20 

1930 

i  35   2 

88  55  21  + 

31 

21 

I931 

35  33 

55  42 

1932 

36   8  3S 

56    2  + 

TABLES. 


215 


POLARIS. 

TABLE  6B. 
For  Greenwich  Mean  Noon. 


Day. 

r 

«2 
IQIO     1930 

<*2 
IQIO    I93O 

Jan.   i 

o.oo 

+  96s  +iois 

+  40"  +40" 

21 

0.06 

74    77 

40     40 

Feb.  10 

O.II 

52    S3 

38     38 

Mar.  2 

0.16 

33    32 

33    34 

22 

O.22 

21     17 

27    28 

Apr.  ii 

0.27 

17    ii 

20     21 

May  i 

°-33 

20      14 

13    14 

21 

0.38 

31     24 

7    8 

June  10 

o-44 

46    41 

2    3 

30 

0.49 

65    62 

0      I 

July  20 

o-5S 

86    84 

0      0 

Aug.  9 

0.60 

105    105 

2      2 

29 

0.66 

122     123 

6    6 

Sept.  1  8 

0.71 

134     137 

12     II 

Oct.  8 

0.77 

140     145 

18    17 

28 

0.82 

140     147 

25    24 

Nov.  17 

0.88 

134     140 

3i    3° 

Dec.  7 

°-93 

121     127 

36   36 

27 

0.98 

102     107 

40    39 

§87- 


R.  A.  =  a1  +  at. 
Dec. 


Interpolate  all  the  quantities  r,  al}  a2, 


INDEX. 


Aberration,  diurnal,  57,  192 
Accurate  determinations,  defined,  6l 

General  principles,  141 
Addition  logarithms,  20,  34 
Adjustments,  of  level,  103 

Of  theodolite,  117,  120 

Of  sextant,  133 

Of  transit,  168,  191 
Almanac,  The,  46 
Altitude,  26 

Reduction  of,  56 
American  E*phemeris,  46 
Angles,  computation  of,  16 
Apparent  solar  time,  35,  39 
Approximate  determinations,  6l 
Approximate  formulae,  9 

Numerical  limits  for,  IO 
Artificial  horizon,  136 
Astronomical  triangle,  31 
Azimuth,  defined,  26 

Computation  of,  65.  152 
Azimuth  determination,  from  sun,  28, 
63,  67 

From  Polaris,  69 

From  star  at  elongation,  86,  89 

From  two  stars,  91,  96 

With  theodolite.  149,  156 

With  transit,  187,  193 
Azimuth  star,  180 

Barometer,  reduction  of,  52 
Bibliography,  196 

Celestial  sphere,  22 
Chronograph,  181 
Chronometer,  care  of,  138 

Correction,  44 

Rate,  45 

Beat,  138 

Precepts  for  use  of,  139 

Comparisons,  139 
Circle  readings,  errors  of,  125 
Circummeridian  altitudes,  79 

Graphical  treatment  of,  80 


Circumpolar  stars,  47 
Clock  stars,  180 
Colatitude,  29 
Collimation,  120 

Elimination  of,  12 1 

Factor,  176 

Mechanical  determination  of,  191 
Coordinates,  systems  of,  24 

Their  uses,  27 

Mutual  relations,  28 

Transformation  of,  30 
Crelle,  multiplication  table,  21 

Day,  35 

Declination,  26 

Determinations  of  azimuth,  latitude, 

time,  60 

Differential  coefficients,  204,  207 
Dip  of  Horizon,  49 
Diurnal  aberration.  57,  192 

Elongation,  defined,  86 

Formulae  for,  86 

Azimuth  determination  at,  88 
Engineer's  transit,  in 
Equator,  celestial,  23  f  26 
Equation  of  time,  40 
Eye  and  ear  observing,  181 

Precision  of,  182 
Equinox,  vernal,  24 

Gradienter,  158 
Calibration  of,  161 
Value  of  a  revolution,  165,  167 

Horizon,  defined,  23 

Dip  of,  49 
Hour  angle,  26 

Of  Polaris,  71 


Index  correction,  131,  135 
Inequality  of  pivots,  no 


217 


218 


INDEX. 


Latitude,  29 

By  meridian  altitude,  61 

By  circummeridian  altitudes,  79 

By  zenith  telescope,  155,  167 
Least  Squares,  178 
Level,  spirit,  99 

Errors  of,  96 

Corrections,  115,  152,  159 
Logarithmic  computation,  5,  12,  1 6 

Accuracy  of,  18 

Tables,  20 
Longitude  and  time,  37 

Magnitudes,  stellar,  47 
Mayer's  equation,  173 
Mean  solar  time,  35,  39 
Meridian,  23,  26 
Micrometer,  calibration  of,  161 

Nadir,  22 

Negative  sign  for  logarithms,  4 
Noon,  36 

Numerical  solution  of  triangle,  5 
Computations,  12,  16 

Observing  list,  142,  147 
Orientation,  60,  70 

Tables  for  Polaris,  211,   214 

Theory  of  tables,  70 

Parallactic  angle,  32 
Parallax,  53 

Personal  equation,  182,  183 
Pivots,  inequality  of,  1 10 
Polaris,  orientation  by,  70 
Poles,  celestial,  22,  26 
Prime  vertical,  23 

Radians,  9,  II 
Reduction  to  meridian, 

With  given  hour  angle,  8l 

With  given  azimuth,  83 

For  zenith  telescope,  161 
Refraction,  nature  of,  50 

Formulae  for,  51 

Coefficients  for  zenith  telescope,  160 
Repetitions,  method  of,  126 

Influence  of  axis  error,  128 
Repsold  Transit  Micrometer,  184. 
Reversal  of  instrument,  113 

Effect  of,  122,  184 
Right  ascension,  26 
Rough  determinations,  defined,  60 

Schedule  for  computation,  13 
Semi-diameter,  52 


Sextant,  129 

Adjustments  of,  132,  133 

Eccentricity  of,  135 

Precepts  for  use  of,  137 
Sidereal  time,  30,  38 

Conversion  of,  40,  43 

Mean  noon,  42 
Sidereal  chronometer,  45 
Solar  time,  35,  38 

Conversion  of,  40,  43 
Spherical  trigonometry,  I 

Fundamental  formulae  of,  4 

Derived  formulae,  8 

Right-angled  triangles,  9 
Spirit-level,  99 

Value  of  a  division,  100,  105 

Theory  of,  101 

Adjustment  of,  103 

Precepts  for  use  01,  104 
Stars,  coordinates  of,  47 

Visibility  in  telescope;,  48,  69 
Subtraction  logarithms,  34 
Sub  polo,  166,  174 
Sun's   altitude,  refraction    correction, 

-  63 

Observed  by  projection,  64 

Tables.     Introduction  to,  199 
Theodolite,  in 

Theory  of,  117 

Errors  of  adjustment,  117,  I2O 

Precepts  for  use  of,  128 
Time,  different  systems,  35 

Determination  of,  60 

From  altitudes,  63,  84 

Meridian  transits,  68,  74 

Two  stars,  91,  96 

Equal  altitudes,  142,  146 

Transit  instrument,  177,  184 

Precision  of  determination,  182 
Transit  factors,  173 

Signs  of,  175 
Transit  instrument,  168 

Adjustment  of,  169 

Theory  of,  170 
Trigonometric  functions,  15 

Vernal  equinox,  24,  26 
Vertical,  22 

Plane,  23 

Circle,  23 

Coordinate,  24 

Zenith,  22,  26 

Zenith  distance,  determination  o£   112 

Zenith  telescope,  157 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 
BERKELEY 

Return  to  desk  from  which  borrowed. 
This  book  is  DUE  on  the  last  date  stamped  below. 


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ONDMY   LIBRARY 


LD  21-100ra-ll,'49(B7146sl6)476 


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UNIVERSITY  OF  CALIFORNIA  LIBRARY 
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